1. Leaving the old wires in place just makes for more wires cluttering up a limited space. I'd either make sure that there're not usable by checking them individually. If you can put power or a meter on one end and confirm continuity I'd use it. If not, if the reason for no continuity is hidden or not reachable, I'd use the old wire to pull through a new wire.
2. Begin at back or front, not sure it makes much difference.
3. 25' should be more than enough length but weaving wiring in and around items always uses more than you think it will.
4. I've always soldered and shrink wrapped all my work on boats and cars for many years and never had a problem. Not everyone is a fan of soldered connections but they work for me
Personally I'd keep my trailer power consumption isolated from my car battery
. I charge my batteries from the car but then I flip a switch and isolate the car battery
from the trailer.
If your only goal is LED lighting
I'd suggest having an inexpensive motorcycle battery as your power source and a small solar
cell to keep it topped up. You might not even require a wire to your car battery but if you do I'd suggest putting a switch or isolator on it to prevent the unhappiness of a dead car battery