Can I use the stock transmission?
Absolutely! and they work great if they've been well maintained. BUT if you're building a race car I wouldn't recommend it (the occasional stop light acceleration run is fine, and hard NOT to do). If you're building an adventure vehicle to take you to a distant trailhead then you'll be fine with your VW transmission, Automatic or manual (I have an auto in my personal bus and absolutely love it).
The stock VW transmission gearing is preferable in mountainous terrain and to performance enthusiasts, while interstate warriors might prefer to re-gear or use a Freeway Flyer transmission offered by many vendors. If you’re not sure about your gearing preference, it makes sense to talk with a VW transmission shop or use the stock gearing to gauge your driving habits and use, as switching out your transmission is simple compared to the engine swap. In short: if you're a blue-highway traveler (think Travels with Charley) - keep it stock. If you're an interstate madman - regear.
Standard use = standard clutch replacement equipment (have had good luck with Sachs and LUK). Strengthened/aftermarket transmission = performance clutch/pressure plate, or standard pressure plate if you don't want to work your left leg so hard when shifting.
Auto Trans Coolers needed?
On Vanagons, yes, go with an aftermarket cooler. On bus and other air-cooled autos, I wish there was an aftermarket cooler but I haven't found one yet.
"Mr. Gas" is a great resource for transmission questions - reach out to him here: https://mrgasrebuilds.com