1) Is the same kind proof of relationship needed (i.e. Facebook messages, Whatsapp, etc.) that I used when I applied for the spousal visa?
No, because for the spousal visa, you had to prove regular communication while you lived APART from each other.
Now that you live together, you don't need emails/messages to prove you communicate, because you see each other every day and live in the same house.
So, for FLR(M), you need to provide evidence of living together in the same house as each other for the last 2 years. The evidence you need to provide is listed clearly in Section 10 of the FLR(M) form - namely 6 items of official correspondence in each of your names (either joint or separately addressed), evenly spaced over the 2 years, from at least 3 different sources
See pages 63-64 of the FLR(M) form:
https://www.gov.uk/government/publications/application-to-extend-stay-in-the-uk-as-a-partner-form-flrm2) The application asks whether we lived permanently since I moved to the UK. I had to live in London for three months (got a job there) while my husband worked out his notice in Bristol. I went back every Friday until he started his new job. For a period of 3 months, does this mean I should say that we didn't live together permanently?
Did you still receive mail at the same address as each other? Were you still living in the same house at each other from Fri-Sun every week?
If so, I would say you were still permanently living together, since your London residency was temporary and you hadn't officially moved there.
However, you could explain in the form that while you maintained your Bristol residence, you travelled to London for work and stayed there 5 days a week for 3 months. As long as you have a valid reason for living in London those 3 months, it shouldn't affect the application.
Edited to add:While I said above that you don't need the same communication evidence, since you were apart in the week for those 3 months, you could also include evidence of phone calls/emails/messages between you while you were in London and he was in Bristol... maybe just a list of a couple of calls/messages per week to show you stayed in contact.
