OK. Another good step forward. The strange dash behaviour, print of codes from an RAC reader, and the one-off start-up do point to one of the following, so let's have a look at them:
1. Earth Leads
Have a look in the Grande Punto section FAQs. There's a guide by Andy Monty about earths and fuse box problems.
2. Fuse Box
See (1)
3. A wiring fault.
Possibly a corroded/loose connector, break in a wire, etc. Could really do with some precise information to narrow it down. You could strike lucky by checking the known villains (rear light connectors, ECU connectors, ECU earth wire, BCU earth wire, and others)
4. Battery or Alternator
I know you've used a different battery, but it is (just) possible that neither your battery or the loan one are giving a stable voltage.
Similarly, I have seen a couple of cases where an alternator seems to be perfectly OK until it gets stressed a bit more than usual. Then it does all kinds of weird things, including flat battery, random electrical problems, etc.
The good news is that it sounds like your gearbox is working fine, and that the troubles you are having are "normal car" rather than "strange gearbox that nobody will touch" ones.
Any moderately competent auto electrician (or even a good local garage) should be able to sort it from here.
Unfortunately, I don't think a generic reader (as nearly always used by this type of place) gives a good enough idea of what's really going on, except for really simple problems.
Where are you based?
A local forum member with
MultiECUScan could capture the existing (FIAT-specific) codes, clear them, and read any newly-generated codes from a couple of start-up attempts. That would give any auto-electrician or local garage a flying start on fixing it.
PS: I'm not criticising the RAC (or AA etc.) info, but I have helped a few people in the past who have been chasing errors from "the breakdown guy" which have not pointed to the real problem, and I'm wary of taking them at face value. The codes read are real codes, but the code descriptions vary in quality/accuracy.