Many insurance companies are allowing customers to postpone their monthly payment.  

"If" you have to go down that road, explain that unfortunately the eviction (while postponed) will still show up on their public record and you will still file a second cause for any past due rents and damages once court resumes.

Her not paying on time resulted in postponed detection of the leak, which I have received counsel on.

You could offer to fix some or all of it, but advise them that the process of them moving in will delay and we will have to postpone the lease to make sure that the items are corrected before they move in. 

I don’t have anything in an email/text about postponement, that all happened via phone. 

Court date is postponed and my hard money lender no longer wants to lend on it

But I don't think that I can postpone the depreciation recapture unless I do a full fledged 1031---an installment sale alone pursuant to IRC 453 will NOT help with that IMHO.The other question concerns the status of (1) "monetized installment sale" or (2) "structured installment sale" or the like. 

The other thing is agents will try and postpone a trustee sale by showing offers so they list it to get an offer to buy some time.

Is this enough to postpone buying the first rental until weather warms up or is this difficulty not really an issue?

There are always excuses postponing the work and not meeting the deadline.