Will 1 loose earth wire in 1 plug affect the house's earth?

No, the 1 loose wire going to your boiler will affect that circuits ability to cut out in the event of a fault etc. but will not affect the installations supply earth or the other circuits protective conductors (earths) n other circuits. nor will it affect that circuits earth from the consumer unit up to the loose wire, for example, if you chopped through the cable between the consumer unit and the fault it would still operate the the MCB/Fuse in the consumer unit using the earth as a fault path as designed. If you chopped through after the fault you could not rely on the earth to do its job (in reality the neutral would provide a fault path but this is one of many scenarios were it wouldn't). Get is checked by a spark, we have a tool called an earth fault loop impedance tester, it will make sure all is ok. Remember if there is no earth to a faulty appliance and the casing has become live its just waiting for you to touch it to give it a path to earth.

Interesting statement by the boiler fitter, why would he be testing all the plugs? is he a spark as well, doesn't sound like it! You need main protective bonding to earth your water and gas pipes if they enter from the ground and are metal, you need supplementary bonding if you have any extraneous metal parts, the boiler should be on a 3a switched fused spur .... let me know if you need any more advice.

Camm no. You bond the gas and water because they are earth. In the event of a fault you want them at same potential as fault. And you also main protective bond extraneous parts eg your gas and water etc etc. Not supplementary.