How to treat pointlike objects in General Relativity? In general relativity we usually treat falling bodies and most small objects as pointlike. It is then enough to solve the geodesic equation in order to predict their motion.
However, it appears to me that a pointlike object in General Relativity should always give rise to a Black Hole. Is that the case? And if so, how is that compatible with our treatment?
 A: First, generally there is no trouble, at least in principle, if one considers point-like sources of fields other than gravitational (such as Maxwell fields) described by linear equations on a fixed curved background spacetimes. Then the fields sourced by point charges are treated as “test fields”, which means that they do not modify background spacetime. We then could find Green functions for the field in this background and obtain the field from the charge by integrating along its worldline. There are some interesting effects arising in such situations. For instance, Green functions even for massless fields would generally be nonzero inside the light-cones (so-called “tails”), so we can say e.g. that electromagnetic field in curved spacetimes propagates not just with the speed of light, but with all the speeds equal or slower than the speed of light.
More trouble arise when one tries to consider the gravitational field of point-like objects. Then, the non-linearity of Einstein field equations poses serious difficulties for mathematically consistent treatment of the problem.
Keep in mind that in general relativity one does not solve the equations of motion for the sources of gravitational field separately from the field itself, instead the evolution of the sources follows from the solution of EFEs. But if we want to describe the motion of point-like object, how then we even could define the “trajectory”  of this motion? If the body is a small black hole, it would seriously modify background spacetime in such a way that one could not recover the worldline on the background spacetime in an unambiguous way.
There are some approaches to deal with this difficulties. We could consider linearized equations of gravitational field on a fixed curved background. But we then could only obtain the first terms of various corrections. And it is difficult to “disentangle” background from perturbations in nonlinear equations.
Mathematically sound is the approach first outlined in this paper:

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*Gralla, S.E. and Wald, R.M., A Rigorous Derivation of Gravitational Self-force, Class. Quantum Grav., 25, 205009, (2008). doi:10.1088/0264-9381/25/20/205009, arXiv:0806.3293.

Here the authors consider a family of metrics $g_{ab}(\lambda)$ smoothly depending on a parameter $\lambda$ where we have the body (or a black hole) of finite size and mass that scales to zero in a self-similar manner with both body size and its mass proportional to $\lambda$. As $\lambda\to 0$ the  body shrinks to point particle worldline $\gamma$, which must be the geodesic of the metric $g_{ab}(\lambda=0)$, which is the background spacetime. One could then consider corrections to the motion of the body and corrections to the metric (which would represent the gravitational field of the body). In principle, one could consider perturbations beyond linear in $\lambda$ order and include more and more features of the body (such as internal angular momentum and higher multipole moments).
Another approach, the method of matching asymptotic expansions, is to consider two zones: the “far field” region, where the perturbation for the background spacetime from the body is small. Gravitational influence from the body on the background in this region could be described by a discrete set of parameters (such as its mass and mass-current multipole moments). Near the body we have the “body region” where the body could seriously modify the metric but the background gravitational fields are assumed to be slow varying and their influence on the body and its motion can be treated as a perturbation. Those two expansions are matched in an intermediate buffer region, where both approximations are assumed to hold: buffer region is far enough from the body to be in its far field yet close enough to be in the body's immediate vicinity by the scales of background spacetime (See figure, schematically illustrating this method using the extreme mass ratio inspiral (EMRI)). Presently this method is the most rigorous/complete approach, thanks to user mmeent for pointing that out.

For more details and further references see the two review papers:

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*Poisson, E., Pound, A., & Vega, I. (2011). The motion of point particles in curved spacetime. Living Reviews in Relativity, 14(1), 7, doi:10.12942/lrr-2011-7.


*Barack, L., & Pound, A. (2018). Self-force and radiation reaction in general relativity. Reports on Progress in Physics, 82(1), 016904, doi:10.1088/1361-6633/aae552, arXiv:1805.10385.
A: Even an extended body has a metric that appears to be that of a black hole far enough from the body.  For example, the Schwarzschild interior solution is an exact solution for the interior of on incompressible fluid, which can be matched exactly to the exterior Schwarzschild solution (i.e. the Schwarzschild black hole solution) at the edge of the fluid.
So if your particle's mass is very small, then its Schwarzschild radius is very small, and unless you can get very close to the particle, you cannot really tell the difference for most purposes between this and a "not-quite-point-like" object has a very small spatial extent.  If your point particle has a large mass, then you probably do want to think of it as a black hole anyway.  If you somehow have a situation that's between these extremes, then you probably need a theory of quantum gravity, which, of course, we don't yet have in consistent form.
