Naruto Shippuuden Episode 163

[Buffalobiian]

I'm also wondering why he can throw Rasenshuriken now. Is it just because he's be able to train it more because Sage Jutsu lets him use it without ruining his arm, or does Sage Jutsu itself allow him to throw it?

If it's the latter, why exactly does it do that? Does having a bunch of Nature chakra just give it the energy it needs to be sustained over a distance?

The Nature chakra allows him to contain the Rasengan until it hits.

Basically, what I'm asking is, when Pain says "Tendo's power won't return for quite some time.", is he talking about his ability to repair the puppet Pain again, or is he talking about his Almighty Push jutsu.

He's talking about Almighty Push. Tendo is basically the "name" of the Yahiko-Pain. That Pain can't use any of his powers for some time, and it so happens his only Power is Almight Push.

Continuing the RasenShuriken talk, the Nature chakra allows him to throw it, so he does, avoiding any damage to his arm. If sustaining the jutsu caused damage, he would have found out during training already.

It's the fact that he's detonating a hand-grenade at point blank that inflicts weaker, but definite, damage to himself.
Now that he can throw it, the problem's solved, only now it actually has a minimum range.

[LunaticCat]

Continuing the RasenShuriken talk, the Nature chakra allows him to throw it, so he does, avoiding any damage to his arm. If sustaining the jutsu caused damage, he would have found out during training already.
(...)
Now that he can throw it, the problem's solved, only now it actually has a minimum range.

that's what I thought, too. It seems to be the most plausile explanation