Great place to spend a couple days, Gila is my preferred stomping grounds when I get out, tons of trail, variety, water, isolation, and great weather.
I think you'll be wise to stick with the West Fork and Middle Fork in late Oct. and Nov., it's all paved road up to the Gila Ranger Station, from there you can jump onto the West or Middle Fork as both waters converge there. Parking is free in that location, you should expect about 1.5 hr drive from Silver City, NM due to all of the slow and twisting roads taking you in. Don't worry about snow pack at trailheads, really don't worry much about snow in Oct. up there, it's not too common in the lower mesas and the Forks in terms of measurable accumulation. The Mogollon Range on the western half of the Gila are more susceptible to snowfall being higher up but are also doable as well, just be prepared for chilly evenings and a longer drive to the trailheads.
Fall in southern New Mexico is dry, precipitation isn't very prevalent; cool sunny days followed with cold crisp evenings. I'd expect to plan for temps in the 20's at the lowest, but definitely below freezing. Daytime temps somewhere in the 50's.
Gear wise, a poncho tarp is sufficient, unless of course the forecast indicates otherwise, but I've done 3 seasons of backpacking in the Gila throughout the wilderness area with nothing more than a solo tarp and a bivy and rarely find I need to pitch the tarp. With the exception of the monsoon season, days and evening are dry and planning for weather is fairly easy and predictable in the fall.
Call the Gila Ranger Station for current flow ratings, but typically, fall flow along the Forks is very light, as the monsoons are long gone, and no measurable amount of snow has fallen by then. Flow rates are really only a concern in the spring when snow run off makes for tricky crossings. Be prepared for over 100 crossings if you plan on going up far along the West or Middle. The high mesas are excellent bypass options if that doesn't sound fun to you.
Here's a trip report I did last spring on a trip taken along the West Fork and Middle Fork, might give you an idea of what you'll see:
Gila River-West and Middle Fork
Best of luck!