Bike fit / correct riding position

Don't need a gym for core work; British Cycling/Team Sky core workouts are

planks
side planks
squats
split squats
windscreen wipers
'proper' press ups

no weights or kit needed except a floor

If you are a BC memeber like what I are there's loads of exercise/stretching/TT warmups etc. info available in their Insight Zone.

Totally. Add a swiss ball (mine was a few squid from tesco), and theres loads you can do. Just giving my gym circuits as an example.

I do a few kettle bell exercises too. That might be overkill though.