Your question isn't dumb at all; it's pretty fundamental to General Relativity.

There are local and global spacetime symmetries.

Locally, the Poincaré group is the symmetry group of spacetime. In at least an infinitesimal neighbourhood around any point in spacetime, that gives you three translations, three boosts, and three rotations about the the three axes with the common sign in the metric signature, and then one translation around the axis with the opposite sign, for ten degrees of freedom.

The "at least" part is important. It is true in general curved spacetime. But you can get Poincaré as the local symmetry across larger regions of spacetime. As the clearest example, Special Relativity is the theory of flat spacetime, and flat spacetime is the special case where Poincaré invariance is true between any two points in the spacetime. Such a spacetime is flat, and intervals are described by the Minkowski metric.

However, general curved spacetimes usually break Poincaré invariance between distant points.

Noether's theorem gives us the conservation of energy from time-translation invariance, conservation of angular momentum from rotation invariance, and conservation of linear momentum from space-translation invariance. In flat spacetime, these conservations are exact and global. In general curved spacetimes they are local: one can fix a boundary around a finite region of sufficiently flat spacetime around a point and confidently predict that all the energy-momentum flowing into that region must eventually flow out (remember that this is a spacetime boundary, so the energy-momentum can stay at the same spacelike coordinates from the "entry" into the spacetime region at t_0 and the "exit" from the spacetime region at t_1).

One can also deal with "sufficiently flat" regions of spacetime in this way, where one can (with a suitable choice of gauge) recover these conservation laws as effective theories. This is exactly what's done with high energy Standard Model experiments in laboratories on Earth, for example: the "lab frame" is effectively flat, so physics can be considered using the Special Relativistic forms of their formal descriptions.

> If space is curved by gravity

Spacetime curvature is essentially gravity in General Relativity. The curvature can squash or strain any of the four axes, and depending on one's choice of axis and system of coordinates, that can indeed result in spatial curvature, however usually physicists prefer to make spatial curvature vanish by choices which produce spatial flatness (and timelike curvature).

Some spacetimes are isotropic. If they are isotropic everywhere then they are also homogeneous. Flat spacetime is isotropic everywhere. That also means the metric is the same throughout space. But there are isotropic spacetimes where the space looks the same whichever direction one looks from a given point. A spatial slice across a single timelike coordinate in a Robertson-Walker spacetime is isotropic, but slices at different time coordinates look different in an expanding Robertson-Walker universe. So here we have isotropy but not homogeneity.

Your intuition that global symmetries (and thus conservation laws) break down in the absence of homogeneity. And indeed, in an expanding spacetime because there is no time-translation invariance, energy is not conserved globally. [1]

Lastly, in general curved spacetimes it starts being difficult to even compare quantities at two distant points p and q that one would want to interpret as energy or momentum or velocity. (One typically sees this put as it being impossible to unambiguously define the relative velocity of two clocks that are not at the same position in spacetime: in short we can have local symmetries being extreeeeemely local and not between points p and q on the manifold except where p == q). Indeed it is often best simply to avoid the concept of inertial frames of reference altogether when one is working with real gravity.

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[1] For the RW case, http://www.preposterousuniverse.com/blog/2010/02/22/energy-i...