If you believe general relativity, and specifically if you believe the assumptions used to derive the FLRW model are justified, then the expansion of the universe is solely the result of the expansion of spacetime. Those "ifs" may seem a bit excessive, but it's important to emphasise that GR is a mathematical model that seems to work when we compare it with experiment. We may discover new things that modify the model. For example the FLRW metric didn't include dark energy, though if dark energy can be described by a cosmological constant the FLRW metric does include it. As the Wiki article mentions, most physicists believe the FLRW metric is a good description of the universe.
To answer your second question requires a bit of background. If you take two non-interacting particles some fixed distance apart and sit back and wait for some significant fraction of a Hubble time you will see the particles moving apart. For the sorts of distances we see every day this effect is tiny; it's only significant at intergalactic distances.
Because the expansion is so small at small distances, it can be counteracted by even the tiniest of forces. This means that if our two test particles are interacting, the interaction will probably completely swamp the expansion. This is why the expansion of the universe isn't making you expand. The forces between the atoms in your body are hugely greater than the expansion effect. Even on galactic scales this is the case. Galaxies don't expand because the gravitational forces between the stars in them overcomes the expansion. It isn't until we get to the scale of galaxy clusters that expansion wins.
The last part of your question is hard to answer without things getting exceedingly technical. In GR you choose some convenient set of co-ordinates to state the metric. The FLRW metric uses co-moving co-ordinates. By definition, in these co-ordinates the time co-ordinate is the same as the proper time, which is an invarient, because the proper time is the time experienced by a freely falling observer. That means that time is not curved. However there is no special distinction between space and time co-ordinates, and there will be other co-ordinate systems in which time is curved.