If you own a province, you continue to own it until another Power occupies it at the conclusion of a fall turn (including retreats).

90kicvesb is correct. If you end up with a unit this Autumn turn in a (neutral) centre that you did not own before, you will take control. After Autumn you will be allowed to build the surplus in one of your home centres (the once you started with) that's unoccupied.

Assuming the described moves happen on an Autumn phase and the bounced territory is one of your home supply centers, then yes, it will be available for a build.

Hi, I'm a new player and I have a question which I hope you could help me with.

Say I have a territory with a build on it, occupied by one of my units. I then issue a move order to a neighbour unoccupied territory, whilst sending a new unit to occupy the first territory. My enemy though has issued an attack order on my original territory. Situation then ends up with my first unit moves to the new territory, and the second unit bounces with the unit of my enemy on the first territory, leaving it empty.

My question is this, in this situation do I still have control of the first territory and use its build/build new unit there?