Administrative Questions

Camping and good connectivity to work online may be a challenge. Any place that offers wifi thru out their park usually are not very reliable, thus cannot be counted on to get much if any work done.

As to your question 4 - you didnt post the type of your rig, types of places you want to camp in and how long have you been traveling part time. There are 1000 things and without narrowing it down hard to tell. One big item many fail to think through is self recovery off road, roadside assistance won't help here. But since you have a jeep you probably already got recovery boards, shovel, a winch, a strap just make sure boards and strap are rated for the weight of your rig. Also if your rig doesnt have a ladder installed you need a collapsible ladder.

There are lots of unisex bathrooms and showers in the US, in anything from KOAs to government run campgrounds. You have to look out for single occupancy "family" type facilities with door to the outside. There are so many you just need to call ahead and confirm this on case by case basis. May be not mention unisex just ask for the above family type bathrooms where entrance is straight from the outside. Use photos in online reviews to scope out bathroom situation too. With forest service and blm campgrounds, its mostly going to be pit toilets, which are all single occupancy and often dont have M/F parts, if you use either no one cares. If you learn how to clean up the old way you only need a real shower once in 2-3 weeks...and then, truck stop showers are almost always unisex, just like many other paid showers in marinas, laundromats, etc. If you cant clean up inside your rig, you need to get pop up shower tent to decouple from shower dependency, plus a garden sprayer or a bucket with a mug and wash cloth. But seems like you have a shower and if you are not using it for storage a folding bucket setup will get you through 3 weeks without a paid shower easy. 

Particularly if coming from a high tax state make sure to cut off as much as possible, your links to your prior state. If you have doctors, bank and other links still in the prior state or you spend a lot of time in your prior state, they may decide you didn't really relocate and will send you a tax bill and it's effectively guilty until proven innocent.

5) Never seen a unisex bathhouse in N. America. On rare occasions, we have found bathroom suites. A row of rooms each with toilet, sink and shower. You also do occasionally see the showers accessed from outside doors with 1 door per shower. Never been much of a concern for us so no idea how to search them out.

7) I won't go so far as to say it's impossible but remote work really works best if you have a skill set and contacts and you just transfer your work from in person to remote. If you are willing to stay in a location for an extended period of time, you can do work camping (usually free campsite in exchange for some work around the campground). Doesn't typically pay well but if you need it to make full timing work....