Sorry about your multiple misfortunes. From the sounds of things, it probably wouldn't have been a good day to buy a lottery ticket, visit a casino or bet on the ponies either. If it's any consolation, you don't have a corner on that market. I have a lot of days like that myself.
Back in 2008, I repaired a cut in my rubber roof (also the result of a falling branch) with Eternabond and never had a problem with it. Unfortunately, I can't tell you anything about the lap sealant. It's been so long, I only remember cleaning the damaged area with something before I applied the Eternabond tape, but with what, I don't know.
If you hate the look of the patches when you look at your roof from inside your house... then don't look at them. First of all, who really cares? It's only a roof. It's not anywhere near as unsightly as having a big embarrassing dent in the trailer from backing into something that everyone and their mother is going to ask you about. Secondly, no one can see patches anyway unless they're higher than the roofline; and how often does that happen? Don't think of them as patches; consider them to be "battle scars".
Can't find what you're looking on JOF? Try:Jayco Owners Forum Custom Google Search
TV: 2009 GMC Sierra 2500HD | Crew Cab | Std. Box | 4WD | Duramax/Allison
RV: 2000 Jayco Eagle 266 | FBS | TT -- 1986 Coleman Laramie pop-up -- Still in the family!!!