Get that hot water washing machine valve capped. Red handle doesn't appear to be fully off, seen a couple of rather large floods caused by someone knocking the lever with something whilst jamming it in an under sink cupboard.
You've already messed about where there's gas involved!
Sometimes a little knowledge is more dangerous than no knowledge at all.
Cut out blockage as above and replace.
Insist on having system flushed out prior to having new boiler fitted to prevent the same thing happening in future.
Rule number 1, never let builders do anything other than build!
Yes it should be tanked underneath, eventually water will get through cracks in grout etc under the tiles affecting the substrate below and they will become loose.
There will be a 'dead' leg on your hot water supply somewhere in your property. Opening and closing a cold tap quickly ( usually happens more on 1/4 turn taps ) compresses the air then releases it resulting in the boiler recognising what it thinks is a hot water demand.
To rectify remove the...
Sounds like you have a dirty system and the boilers main heat exchanger has become partially blocked/restricted.
Are there any noises before the boiler goes off?