Anybody got any ideas on this one?

I have a really annoying intermittant fault, where I turn the ignition on
(not trying to start the engine) and all warning lights on the dash
flicker with an annoying clicking sound coming from the passenger side
rear of the engine bay (opened her up, appears to be a relay next to large
hose). When the car is in this state and try to turn the engine over, I get
one click from the starter then nothing. I've previously had a fault (which
went away) where the starter would appear to stick, again this only
happened a few times.

I've taken the car to the garage and they claim that its not the correct
battery for the car (thinking why is it only presenting problems now - and
still intermittent, if this is the case)?

Also, sometimes when driving - the oil pressure guage will plummet when
accelerating then starts moving erratically from 0 to 8 bar!?

A couple of days before the ignition problems started occurring again, I
had changed the engine oil and filter - along with a new set of plugs.
Silly oversight but managed to close the bonnet while still having the oil
filler cap off (noticed smell of burning oil 40 miles down the road then
sorted). there wasn't much oil loss, but I topped up to MAX anyway and
thought that'd be the end of it. Is it possible that any engine damage may
have been caused due to lack of pressure in the rocker cover?

Any help appreciated.