1) Since you can only place units where they will be in supply and a unit needs to be in your home base in order to provide supply, the answer is yes (exception: paratroops can be placed out of supply).
Just confirming what's already been said. A unit is considered supplied if and only if it is in its home base or can draw supply from a line of supplied units that extends to the base. This means that if the base is empty there's nowhere (other than the base) that a unit can be played with supply, which in turn means that Paratroopers are the only unit that can be played outside of home base when home base is empty. This also applies if a supply chain is broken up by destroying a unit in the chain; other units may become unsupplied, and unless that chain (or another one) is re-established you can't use those unsupplied units to enable new deployments.
I'd also point out that if your opponent has a unit in play you can use an Air Strike as your play rather than playing a unit.