How much cargo weight for moving across country in my car?

That's a big issue with crossover/ small SUVs that noone ever discusses. Four 200 lb adults and almost maxes them out without any luggage yet. Do not exceed 850 lbs. The vehicle becomes unstable and can even be dangerous beyond that. 

850 or less.  Put the heavy stuff on the floor & lighter stuff higher. Books & magazines R heavy. Honda's cannot carry a heavy load. Shocks only stop the bouncing of the vehicle. It cannot carry more. It will not corner very well or have same acceleration 2 climb hills or take off speed. Hire a U Haul truck.  Maybe hire a guy to drive the truck.

Obviously you're way inexperienced behind the wheel, but kudos for asking. Even if you distribute the weight evenly accordinbg to the axle weights listed for your vehicle, The main issues will be the condition of your shock-absoirbers, what load-rating and tread your tires have, and whether the steering system and brakes are in good shape. I'd install heavy-duty adjustable shocks and good quality all-weather load-range B tires, and have the brakes and front-end checked for wear. Be careful with following distances at all speeds because braking distance will be a lot greater due to higher rolling mass of the loaded car. If you don't take these precautions, chances are that you're an accident waiting to happen. 

Just to show you what can be done, a few years ago I drove a 1970's Datsun mini pickup truck 4,000 miles from Alaska to Arizona that was over-loaded by 700 pounds rated load 1000, actual load 1700. And that was driving over roads a LOT worse than what you'll be driving. Before going I put helper springs and adjustable air shocks on the back, heavy duty shocks on the front, high load rating tires all the way around, and made sure the brakes, brake lines and steering were in good shape. It was a little slow going uphill but it performed very well. 

One other thing, you can always SHIP some of your stuff. 

First of all, be aware that carrying the max load necessitates increasing tire pressure. Just how much I am not sure.

Secondly make sure the tires are not rubbing on the lip around the wheel cut outs.

Thirdly remember the car will not handle the same way, this includes stopping!

Forth, don't plan on driving at the speed limit and don't plan on making up time going down the mountains! Reduce the speed and be aware that going down hill can be more of a hazard then going up one. If you loose your brakes going up a mountain, the climb will slow you down. But loose your brakes going down hill only results in going faster.

The reason the car has a rated weight capacity is because of the suspension and tires on the car. The Civic can pull a small trailer, so it is not how much it can pull. It might not top the grades at top speed and if it is an automatic there is a chance of damaging the transmission. Don't over heat it! So what might take four days with out a load could end up taking eight or nine days because there are a lot of steep grades out there. Having to go only 55 mph. instead of 65 or 70 can add up real fast when it comes to driving time.

I used a Miata 1.6 to pull a fifteen foot boat 1500 miles, a 79 Subaru (FWD) to  pull a trailer with 1000 lbs plus the trailer about the same distance (during a blizzard) and used to pile stuff in to and on to a 72 Corolla all to frequently. The cars held together provided I adjusted speeds and tire pressure accordingly.

Your weight calculation also has to allow for the weight of a full gas tank.

You also didn’t mention the altitude of the mountain roads, but if there are steep gradients you will likely need to reduce the total weight limit by at least 30% (less if it’s a diesel): steep gradients, and especially at altitude not only reduce engine power output but put a huge extra load on your brakes as you descend whilst using them, so use engine braking as much as you can.

Unfortunately your car was never designed to lug heavy loads. 

For weight distribution just spread it evenly around the car, but make sure that nothing can break free and roll into your footwell and that your mirrors and vision out of the car are unimpaired. Too much weight at the back will make the steering hazardously light and imprecise and downright dangerous in wet or icy conditions. Although your car is AWD it’s far more accurate to describe it as FWD with a little bit of RWD assistance when needed. So an overloaded rear with light steering will also cause a loss of FWD traction.

It’s an ambitious drive with that load. In your position I’d consider contacting shipping companies to see what it would cost if you sent your heavier items and stuff you don’t yet need (summer clothing) by using standard palletised crates. If you do that make sure that the shipper knows there is no fork lift truck at the destination so that they only choose trucks which carry their own lifts. Doing it that way is not just safer but means you have no worries about your car being broken into when you sleep. With far more space in the car you could even consider saving money by sleeping in the car as you can then recline one of the front seats.

Weight WILL change the handling of a vehicle -- deal with it.

More weight, more stress on the vehicle, harder to stop -- reduce your speed.

You can overload your vehicle, it will not destroy your vehicle.