Steam – Is it possible to log your Steam account into two computers at once

Yes, games linked to your Steam account can be installed / uninstalled at will.

There are some rare exceptions where a game has additional DRM that limits the number of times you can install - but this should always be noted on the game's store page (the first example I've found is Dead Space 2, check the wall of capitalised text carefully to see what I mean) and there is often a way to "reclaim" those installs before you uninstall the game. So check the Store page for each game before you uninstall if this is a concern.

When you want to reinstall:
On your game list, if you switch to the "All Games" view you will find all the games you have linked to your account. Those that are not currently installed will be greyed out, and from here you can select for them to be downloaded and installed.

It doesn't look like it. I just tried installing Bioshock Infinite from my phone, and it didn't give me an option of which library (I have 2) to install to. It looks like it will install to the default (i.e. the first) library.

If that isn't the library you want to install it to, then you can always move the game after it finishes installing.

It doesn't seem like you're able to change which library is the default either. If you want to change which library is the default, you'll have to reinstall Steam onto the drive where you want your default library.