Objects in Physics as a mathematician would see them I'm a mathematician with hardly any knowledge of physics. Before I start reading volumes of physics books, I have a few questions that have been bugging me and that will help me start reading physics.
Let's forget Quantum Mechanics for the purpose of the discussion, and focus on Relativity. I always fail to understand what the objects are and how to relate them intuitively to perceivable space and time. What are the objects?
Here's what I mean: we begin with spacetime being a pseudo-Riemannian manifold with some metric (is this metric assumed to be locally Lorentzian?). For each point in this manifold, there should be some parameters, right? Like whether there's a particle there, what particle is it (information that includes, for example, mass), is there a magnetic field there and so forth. Together, this pseudo-Riemannian manifold, with a pre-specified metric, and with a set of parameters for each point, is what described the universe in relativity, right? What precisely are those parameters in General Relativity?
 A: Those set of parameters are usually called matter fields. They contribute to GR on the "right hand side" of the equation 
$$ G_{\mu\nu} = T_{\mu\nu} $$
Since you mentioned that you are a mathematician, let me be slightly more technical and verbose. The space-time is given by a Lorentzian manifold $(M,g)$. The matter-fields should be considered to be a collection $\{\Phi_A\}_{A\in \mathcal{A}}$ where $\mathcal{A}$ is some indexing set, with each individual $\Phi_A$ being a section of some fibre bundle $(E_A,M,\pi_A)$ over $M$. (There are more general possibilities for matter fields, but let us stick to these now.) For example, the scalar field is given by a trivial complex line bundle over $M$, while the usual description of Maxwell's theory of electro-magnetism admits the formulation of the field (the vector potential) as a section of the cotangent bundle $T^*M$. 
The dynamics of the fields are generally prescribed by some equations of motions, and their contribution to gravity is taken to be their contribution to the energy momentum tensor
$$ T_{\mu\nu} = T_{\mu\nu}(\{ \Phi_A\}_{A\in \mathcal{A}}) $$
in such a way that the condition 
$$ \nabla^\mu T_{\mu\nu} = 0 $$
where $\nabla$ is the covariant derivative of the metric $g$ is satisfied. (This is due to that the identity must hold for the left hand side of Einstein's equation according to the contracted Bianchi identity.)
Note that general relativity by itself is a theory of gravity. It doesn't really specify what the matter fields are. It only requires that the matter fields, when they do exist, have dynamics that obey the conservation law given by the divergence condition on the energy-momentum tensor. To get an actual physical model of the world, you will have to come-up with some rules that govern the behaviour of the matter fields. In modern physics, this is generally through some sort of action principle, since the Euler-Lagrange equations will automatically be compatible with the divergence condition above, if you take $T_{\mu\nu}$ to be the Einstein-Hilbert stress-energy generated from the action principle. 
A: In GR, the "objects" (whatever they might be for your problem) need to be mapped to the stress-energy tensor, which the GR field equations relate to the curvature of the manifold. That mapping is not really part of GR though but it is surely part of the everyday application of GR, although it will differ depending on the problem and what level of description of the physical objects is used.
One introductory physical object which is often used as an example in introductory GR texts as far as I know is "dust", to quote Schutz: "'dust' is defined to be a collection of particles, all of which are at rest in some one Lorentz frame". You define a density of the particles in a volume, and the momentum flux component $\alpha$ of the particles across each perpendicular surface of constant $\beta$ in spacetime is one of the components of the stress-energy tensor $T^{\alpha\beta}$. So $T^{00}$ is the energy-components of the dust particles across a surface of constant time, that is, the energy density. 
From there you can keep adding more complex objects like fluids or electromagnetic fields etc. as long as you map them to the stress-energy tensor.
A: I would say, as far as general relativity is concerned, "object" is a fairly vacuous term. What one is really looking at is some perturbation in the stress-energy tensor. Einstein's Equations (without cosmological constant) are $G_{\mu\nu}= \frac{8 \pi G}{c^4} T_{\mu\nu}$, which defines a relation between Einstein curvature, and therefore the metric, and the stress-energy tensor at a given point. I think this tensor is what these "parameters" you are looking for are, as it defines the energy density, the energy flux, the shear stress, and the pressure. 
A: The 'objects' that are studies in relativistic physics are the same as what people are studying in non-relativistic physics.  Consider a few examples:


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*If you are concerned with relativistic mechanics one can consider extended objects like rods of length L and calculate moments of intertia. (Here the object is the rod)

*If you are studying quantum mechanics then you will study the Dirac equation rather than the Schrodenger equation (Here the objects being studies are electrons)

*If you are studying statistical mechanics you may study the equation of state using a gas of photons rather than some non-relativistic gas. (not if object here applies unless we're talking about the collection of photons themselves)
The basic things which are studied are still the same things you'd study in non-relativistic physics.  Nothing really changes there.  What does change is the geometry of the underlying manifolds where the objects of study live.  This usually adds two things regardless of wether you're talking special or general relativity:


*

*Extra terms in the potentials used to describe interactions between the 'objects'

*Extra terms in the equations of motions which arise from the geometry of the space-time the objects live in.
General relativity is still physics so look to physics first to understand what you are trying to study.  The questions being asked are still the same.  I know sometimes it's hard to see that when you pull open a book that says general relativity on it and the discussion revolves around connections, curvature, exterior algebra, and so forth but the objects of study are still there as you correctly assert.  :)  My suggestion is to crack open a basic physics book and then try to find the relativistic analogs of what you see there in more advanced books like Misner, Thorne, and Wheeler's Gravitation or Robert Wald's General Relativity.  They both have very physical approaches to GR and you may have an easier time getting at the 'objects' that way.
A: General relativity is a mathematical theory that generalizes the special theory of relativity and mechanics. It turns out that one can treat the gravitational effects as non-inertial forces, thanks to one of the principles that governs this theory, that is also considered the most important one, namely the principle of equivalence postulated by Einstein. That the metric of space-time must be locally flat, i.e. Minkowskian, is a consequence of this principle. In fact, the principle of equivalence states that in a free falling frame of reference, i.e. a reference where there are no inertial forces, the laws of physics are those of special relativity. As you are a mathematician there is no need for me to recall that if a manifold has a null curvature tensor, then there are global coordinates on the manifold such that the components of the metric tensor, say $g$, are exactly those of the Minkowski metric $\eta = \operatorname{diag}(1,-1,-1,-1)$.
The metric $g$ on the manifold $M$ describing the space-time is not a-priori given, but it is determined by the distribution of matter. Now comes your question on the objects of general relativity. These objects are all sorts of geometrical object that you can construct on a (smooth) manifold. Thus you'll have tensors of any ranks, and even spinors of any rank. What makes them physical objects is just their interpretation. Einstein's field equations
$$\text{Ric}-\frac12\operatorname{Tr}(\text{Ric})g = \chi T,$$
where $\text{Ric}\in T^*M\otimes T^*M$ is the Ricci tensor, $g\in T^*M\otimes T^*M$ the metric on $M$, and $T\in T^*M\otimes T^*M$ the energy-stress tensor, gives you the metic $g$ in terms of the energy-matter distribution $T$ on $M$, that is just a tensor density on $M$.
The mathematical problem of studying the geodesics on a manifold $M$ described by a metric $g$ is the equivalent of the physical problem of determining the motion of a particle that is freely falling in the gravitational field generated by a matter distribution $T$ such that the resulting metric is $g$.
One of the most spectacular prediction of the general theory of relativity is the existence of so-called black holes. Again you find these object by studying the mathematical properties of the solutions of the Einstein's equations and then give them a physical interpretation. There are some tough situation in physics when you can't base your theories on observations and experiments. Therefore you must start with a mathematical theory and develop it as further as you can. Each result you obtain is then physically interpreted. This is quite the situation of general relativity (see, e.g. the problem of the detection of gravitaional waves, or the observation of naked singularities).
A: Classical GR can only study objects which can be modelled by tensor fields (on the Lorentzian manifold of space-time).  The metric, of course, is itelf a tensor field, and the Laws of Nature have to take the form of equating two tensor fields with the same covariance properties, i.e., of the same type.  Now it would be pretty silly if the metric itself didn't get used in the equation, so this puts some limitations on one's search...
If matter is regarded as a kind of continuous distribution, like a density, it can be well modelled by the stress-energy tensor mentioned by the other posters.  Hydrodynamics can be done pretty well too.  Electromagnetism, less well, but something can be done.  To answer your question very directly, then, the only parameters are the coordinates of space and time and the only objects are these tensor fields, and this limits GR so that it cannot do a good job considering quantum effects like wavicles or spin.  Classical particles can be treated by letting the density of matter, considered as a kind of « fluid » (I think that is a better word than powder or dust) of mass-energy, have some singularities, they definitely do not get any kind of parameter of their very own. In this respect, it is very like Newtonian and Eulerian dynamics and hydrodynamics. 
