The Bauer seems to be an American-made copy of the Baby Browning. Some sources on the web seem to feel that it is a very-high-quality piece.
By the way, it's stainless steel. (Are you sure that the grips are fake mother-if-pearl?)
One source prices it at $150.00 to $200.00.
Search on the web for "Bauer Automatic .25," and you will find disassembly instructions, among other things.
Clean it as thoroughly as possible, lubricate it lightly, put it together correctly, and it will probably be safe to fire.
Its magazine holds five rounds, plus one in the chamber.
There is no "cheap" .25 ACP ammunition. It's somewhat hard to find, and it'll be expensive. It's also almost useless, except as a "get the heck offa me" round at very close range.
I dunno nothin' 'bout no Browning Buckmark .22, but I'm sure there are people here who can help you with it.