1. Writing on it "for settlement purposes only" wouldn't be applicable unless you're making a settlement offer. Writing that and then saying sorry could still be presented. I won't get into it with you, but if you acknowledge fault in an accident your insurance can deny you due to admitting fault before clear liability was determined.
I'm not talking what your insurance carrier will do, I'm talking about whether anything would be admissible as evidence. Rule of Evidence 408 provides that a correspondence (written or oral) made for the purposes of settlement is not admissible as evidence to prove liability. If you submit a correspondence inquiring for information for settlement purposes, that's not admissible as evidence. Courts generally broadly enforce Rule 408 because they want parties to work it out without filing suit.
2. Laugh all you want, they are required to share with you their evidence. If you received a demand letter (like it sounds like you did since they asked for $2,000) they should have sent over their compelling evidence as to why you're at fault. If they are going to sue you they will have to file evidence anyway. They aren't going to show up in small claims court and reveal everything for the first time.
They are absolutely not required to send you anything when they send a demand letter. I have been practicing for five years and not once have I sent an exhibit (unless it's an underlying contract) to somebody else with my demand letters.
And they ABSOLUTELY WILL show up at small claims court with evidence for the first time. The civil rules of discovery do not apply to small claims actions. You would have to transfer it to the regular docket, which would create a much more expensive process for all involved.
If a lay person requested my file prior to a small claims hearing, I would not turn it over, period.
3. Who says the GC didn't build a trench or do anything to try and prevent water run off? I don't think this is/was clear based on what I read.
The neighboring property is getting water runoff since this construction began. I'm going to imagine this didn't happen prior to the construction, because the homeowner could have just sued the lot owner/done something a long time ago even with no structure on it.
4. If it's an insurance claim, most recoveries by the attorney are capped. This may not be an insurance claim ultimately, and this also doesn't apply to if the attorney charged a retainer, but it's certainly not meaningless. It could be a family member or a friend who's an attorney who has no intent to do anything more than send a letter.
We're talking about two things here. You're talking about capped fees, i.e., statutory recovery of fees as part of their case. That won't happen here.
But this is clearly an hourly case (or, like you said, a pro bono case). An hourly fee is completely divorced from the underlying result. I have two cases now where my fees are above the amount in controversy -- not because I'm milking clients, but because they have so much money they don't give a shit, and they'll pay whatever to recover what they feel they are owed.
Edit:
Just editing to say your best first step is to either contact your insurance company or contact an attorney if you wish to handle this yourself. Wishing you the best.
We agree on something at least.