"Also, where are dump stations that are not connected to camp grounds? "
A variety of places... highway rest stops, some stores (Cabela's, Bass Pro Shops, etc), local town dump (check with your town hall), etc.
"And, along that same line of question, what is the road etiquette for dumping at camp grounds? "
Call ahead is probably best - so you know if you can or not. Many campgrounds will allow non-campers to dump, BUT there will probably be a fee, sometimes that fee is only $5, sometimes it may be the same as a night's stay or even more. My local state park allows non-campers to dump, but the fee is the same as an overnight stay.
"Do you have to have stayed there? or can you just use their facilities? "
I may have answered this above if you mean just dumping when you say "facilities", if you also mean using their showers, bathrooms, laundry, etc just ask at the office. It will certainly depend on the campground, and probably the mood of whoever happens to be at the front desk at the time you pull in. Some campgrounds will allow it (possibly with a fee), some are very strict. You won't know until you ask... which is why it is usually better to call ahead before you get off the highway and pull into a campground.
With careful use, your camper's black and grey tanks should easily last for 3 days, more like a week when you get used to things. Take "Navy showers" (turn on water to wet down, shut off water, soap up, turn on water to rinse off as quickly as possible, shut off water) - you may be surprised at how little water you actually need for a shower. Paper plates and plastic ware that you can throw out so you reduce dish washing. Lots of ways to reduce water usage as you will find out with experience.
We routinely camp for 4 day weekends (Friday thru Monday) with no hookups and don't run out of water or fill our waste tanks.
"Full hookup" campsites will seem like a big luxury - use all the power you want, and take nice long showers! But... you may find that once you get into the habit of using less, you will do it all the time.
There is tons of information on these forums. Use the search feature. When you post questions, remember to give as much detail as possible to allow people to give better answers.
And see you on the road!