Of course, if the chain/cassette are worn out they should be replaced. If you haven't replaced the chain in 10K miles the chainrings are probably shot too.

There shouldn't be anything wrong with your brakes. But you might want better brake pads. Kool Stop salmon are the best. It's also likely that your brakes aren't setup as well as they could be. The fanciest brakes in the world won't fix poor setup.

You can upgrade whatever you want, Tiagra or 105 will fit that bike. But to me, it would make the most sense to just replace the cassette/chain and busted shifter.