Do you believe in free will?

[oxymoron]

2. There is no mechanism, it is outside of the material realm, just as some would argue numbers and time exist outside of the material realm.

So it's random then. Because numbers and time have mechanism - they're quite deterministic (in that they exhibit regularities which we can write down, known as "mathematics" and "physics"). The specific ontology is irrelevant, however interesting it may be to some.

Further, whilst time and numbers would appear to be independent of matter, brains are not. Again: "mind" is an article of rhetoric, not a scientifically or even philosophically valid term. Since minds are never discussed without brains to host them (or equivalently: this has never been observed, however easily it can be hypothesised), they are contingent upon matter in a way that numbers and time are not (or if they were, would void your argument).

3. The ability for the non-physical conscience to effect the material world is the action of free will.

"Non-physical" = "Magic Pixies".

There's no evidence for a non-physical agent, no need to invoke one, and when you do you just end up with absurdities - like brains being irrelevant to minds.

Does someone who suffers from complete amnesia after a horrific accident maintain their identity?

Depends on what "complete" might constitute. You'd pretty much need to liquidize their entire brain for that to happen. But in general, from real-life case studies, no, they could be unrecognisable to others in terms of personality, and the identity they do construct post-incident would be different to that previous to the accident.

Similarly, if a person is declared dead, then resuscitated with no remarkable changes to their brain functioning, do they retain their identity - even if their temporal continuity has been interrupted by a brief instance of non-existence (death)?

Probably. Memory is encoded in structure, and a brief blip of the power (so to speak) which does not affect that structure then does not affect that memory, by definition. A difference that makes no difference is not a difference.