The relay may be needed because of the buttons instead of the mechanical switch. The main purpose is likely to isolate the Ohms current source - also using it for the mV ranges is more like an extra to keep the noise low.
Yep. It looks like it switches in when the probes are an "output" (Ohms, continuity, diode, etc.)
The the PTC should really be before before the relay, but they put the PTC
after the range switch in their other meters so nothing new there. It probably doesn't matter, if you're doing something incorrect enough to wreck the range switch then that little PTC probably isn't going to save you.
The soldering on the reference does not look that good - it's likely soldered in the other side.
It must be touching
somewhere or the meter wouldn't work.
I'd like to give it a wiggle and see what happens.