Is anything actually 1 meter long (or 1kg of weight)? I believe that no real objects are actually (exactly) 1 meter long, since for something to be 1.00000000... meters long, we would have to have the ability to measure with infinite precision. Obviously, this can be extended to any units of measurement. Am I wrong?
 A: 1 kg is defined as the mass of a particular reference mass in France.  So there is at least one thing that weighs exactly 1 kg.
Furthermore, it may in general be possible to construct an arbitrary number of objects which weigh 1kg exactly. Assuming all Si atoms have the same mass, one could simply define the kilogram to be the mass of a certain number of Si atoms.
A: You're not wrong. However, there used to be an object exactly $1$ meter long until 1960, because a meter was defined to be the length of a certain platinum-iridium rod at certain conditions. Since then, the meter is defined in terms of interferometry, and now it is specifically the distance traversed by light in vacuum within a certain period of time.
Similarly, the kilogram has a prototype whose mass is $1$ kg by definition. There are some proposals to replace that definition, but it hasn't been done yet.
A: In Nature the only exact realization of a number is thru $\mathbb{N}$ i.e. counting . I suspect that they are called 'Naturals' because of that. To all the other numbers we only get aproximations. Even when defined to be the exact length/weight of a certain physical object the measure of it will be within a certain range of values, $\pm$Angstrom. 
Another example: there is not a single realization in the universe that correspond to the number $\pi$. Numbers are menthal constructs.  
How do we measure ? Obtaining a ratio between the amount that is to be measured and the amount of a selected standard. If both amounts change equally, the measure (ratio) keep constant, i.e. the constancy of a measure is not a guarantee that the properties of 'objects' do not change.  
Atomic measures are number counts:  

We will now see that the measures of bodies' properties using atomic
  units are independent of the base quantities and dependent on the
  number of particles or atoms.
An atomic unit of mass is the mass of a certain number of baryons; the
  measure of the mass of a body using atomic units is therefore a number
  proportional to the number of baryons of the body (this is not an
  exact statement but it serves the needs of this work). If the mass of
  baryons changes, so will the mass unit and the mass of the body; the
  measure holds invariant because the number of baryons did not change.
  Therefore, a measure of the mass of a body using atomic units is
  basically a baryon count, holding invariant as long as the number of
  baryons does not change, independently of the eventual change of
  baryons' mass. The same kind of reasoning applies to charge measures.
  In what concerns length measures, the length unit is such that the
  measures of length of isolated bodies hold invariant; this is not the
  way length unit is formally defined, but this is a condition it has to
  obey to be acceptable, in order to fit Einstein's measuring rod or
  reference-body, translated in the time invariance of Bohr radius. So,
  we can say that the atomic length unit is a fixed multiple of the Bohr
  radius; if the latter varies, so will bodies' length and the unit of
  length, holding invariant the measures of bodies' length. Therefore,
  length measures are a way of counting atoms, the measures of the
  length of bodies holding invariant as long as the number of atoms does
  so, for bodies and measuring devices subject to the same conditions. 
The above reasoning shows that the measures of mass, charge and length
  of bodies are independent of the mass and charge of elementary
  particles and of atoms' radii, tracing only the number of particles or
  atoms.

To any measure (ratio) that is not obtained thru direct counting an error margin is present (explicit or implicit) .
A: No, but for other reasons than you might suspect. For two endpoints of an object to be precisely one meter apart, one would have to be able to consistently measure that distance. But in reality, the outcome of any individual measurement is constrained by the Heisenberg inequality, and therefore you cannot consistently measure such a length. If you'd know the precise position of one endpoint, you'd have zero knowledge about it's speed.
Now there are also practical limitations to our ability to measure lengths, and they kick in much earlier (typically ~10-9 or worse; in comparison Planck's constant is ~10-34)
A: The definition of units in the SI are determined by the Bureau International des Poids et Mesures (BIPM). The official statement of the definition of the kilogram is

Le kilogramme est l’unité de masse ; il est égal à la masse du prototype
  international du kilogramme.

The English translation is unofficial, but it states

The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

Thus, as was stated earlier, there is an object exactly one kilogram in mass.  You can find the BIPM brochure that contains this definition here.
