Yes. The spell requires concentration to maintain, and it lets you do certain things "once per round". If you cast it during your round, you do that certain thing in that round. Concentration is needed after, but the standard action is to maintain the spell, not to do the stuff.
Yea, sounds right to me. You gain the ability for immediate use upon casting the spell, but need to make a Concentration check each subsequent round to continue using it. While active, you can do anything normally possible via telekinesis.
For sure. From the reading of the spell, using it to do something is one of those "not an action" actions like knocking an arrow to a bow. Casting the spell is an action, concentrating on the spell is an action, but the "doing something" non-action is intrinsic to the action you have to take anyway to either bring the spell into being or maintain it.