If you have some spare cash an electric shower is normally cheaper to run, but obviously you need to buy and fit one
Do turn the plugs off at the sockets, lots of things use as much power on stand by as they do when working
Both of these are completely wrong. An electric shower is probably the most power hungry appliance in domestic homes and is way more expensive to run than combi heated water. Possibly immersion water too.
From what the OP has said, it sounds like she's been heating water with an immersion heater and not using it, because of the combi (surely not?) although it would be interesting to see what the plumber says (I don't know if a plumber is the right person, but someone from a plumber/heating engineer/electrian(?) will be able to trace the pipes etc from heater to taps to determine what is going on).
On the matter of standby and sockets, items that are plugged in but off, or even on standby, unless they're really old, because of an EU regulation that requires all items manufactured after a certain date a good while ago to use less than 0.5 W on standby. Which is nothing. Stuff costs pence, or a couple of pounds a year on standby.
OP, what you need to concentrate on is what is heating your home, water or food and to a lesser degree fridges etc, halogen/old fashioned light bulbs and possibly anything like a fish tank pump or gaming PCs with high capacity fans.
Phone chargers, TVs on standby and digital clocks is fiddling round the edges.