Plumbhandy

Hi possibly they require a flush of the system or they require levelling up which means adjusting all radiators until they heating up equally

No i would charge a fee for an airlock

I would close of the water to the and replace the tap connector with a new valve, olive with ptfe tape to solve any future problems

I would try fitting a gate valve on the cold water supply to the shower to reduce the cold water pressure to the shower giving it an equal flow of hot and cold to the shower

I don't know the sink in question but i reckon a 32 mm waste clicker or pop up should fit i would require to look at the sink to see what needs done