In 2013 I made an attempt at the Tour Divide. It didn't work out as planned, but I do have some insight regarding the first 1,000 miles or so.
Bike- KISS. I rode on an aluminum hardtail that I had converted to drop bars with Genevale shifters and cable brakes. No particular complaints with the drops, between those and aerobars I had lots of positions, helped stave off the faatigue of staying in one position all the time. Would not ride the same bike (Redline D680, an XC race machine. If I were to go back and ride it again at a touring pace, would choose one of three Surly models, the Karate Monkey, the ECR or any one of their fat bikes. Regardless of brand I would stick with a steel frame (or Ti if you can afford it), rigid fork and the wide tires, at least a 2.5, 3.0 if you can fit it, 4.0+ if you go for a fatty. Good cable operated disk brakes provide plenty of stopping power and are easy to field fix if needed. Would suggest a 1X drivetrain with a fairly small chainring, some of the climbs are grueling.
Training- lots of it, and as similar to ride conditions as possible. At least once a week try to ride your anticipated daily milage on dirt with your full load.
Mindset- you are out there to have fun. Some days will suck, others will be wonderful, enjoy the ride.
Time off- I had a buttload of PTO racked up, YMMV.
Gear- even though you are not racing, keeping it light will make the trip easier and more fun. I wouldn't reccomend a trailer. I was trying to race, so I went ultralight, no stove or cook gear, bare minimum clothing for the anticipated weather. You will need a way to carry plenty of water for the southern end of the route, and really should bring a way to purify water from sketchy sources. Sawyer filters are light, compact and effective. I didn't use a backpack, kept everything on the bike. If you can handle all day rides with a pack, use a hydration pack for water and a few other incidentals, free up space in your other bags for a few more comfort items than I carried. Carry some good rain gear, I got caught in one MF of a storm, bad enough that when i hit pavement again heading in to town (Helena?) I got pulled over by a cop telling me to get off the road, out of the weather, which was exactly what I was trying to do.
Shelter- I used a minimalist bivvy bag. I wouldn't do it again. get a light, compact one or two person tent, it will be worth the minimal extra weight.
ETC- Have fun! I would love to do it again, finish the route at a tourist pace.
If you want a somewhat more manageable route, or can't get enough time for 2700miles, look at the Oregon Timber Trail, 670 miles, 55% singletrack.