Of course they are very much related. In the absence of any gravity pressurisation, the pressure in the hot water system is a direct consequence of the cold water supply pressure.Not a chance. Firstly, the boiler has no involvement with the cold supply to the mixer - they are not related, and there is no reason why they should remain either constant or at the same ratio.
Sure, but that's a totally different problem from what the OP is experiencing and what we have been discussing, and is a problem with any plumbing system if some other 'load' like a loo shares a pipe run with the feed to a shower. The (common) problem which the OP is talking about will/can exist even if both cold and hot feeds to the shower are totally 'dedicated'.Secondly, if the pipe taking cold water to the bathroom also serves the WC, then causing the cistern to fill will without doubt reduce the flow to the bath taps.
Kind Regards, John