Keep in mind that any water you are cooking with is still liquids you are intaking. A pantry full of dehydrated food is an awesome weightsaver in places were you always have water available but in a dry place you need to carry it with you.
Efficiency-wise, I personally found the scout kelly-kettle type stoves ( kelly kettle + pot on top) great for boiling water while cooking another at the same time in the pot on top of the kettle, not much fuel needed. And one way to conserve the water is to rinse the cup you ate in and drink it before washing it
If you were to use the precooked rice with warm water it takes about half hour to bring it back and you still should bring it to a boil after. I know of people that just soak it for a long time and eat it as it is but I think they must be crazy.
Since from your question it seems you think you cook rice boiling, draining and throwing the water out, be aware that many kinds of rice don't work that way.
If you take uncooked Jasmine rice, for example, the ratio is 1 part rice and 1 1/2 part water and all the water is absorbed by the rice (so way less than a liter, 250 ml of water will be even too much for one person). Forget Arborio or similar as they take long time to cook. Precooked rice works the same as it will absorb the water and at 1 to 1 ratio usually and faster cooking. You might want to redo your calculations then and probably find out you can expend the fuel to bring the water to a boil ;). Also Dhara gave you an excellent suggestion in the comments as there are recipes for eating the flakes raw it seems.
Then, if what you mean is the ready to eat kind that comes in a bowl or similar to cook in a microwave or stovetop, that usually involves boiling the unopened container in water for more than just a few minutes. I suppose you could reuse the same water over and over as Aravona mentioned in the comments but then that stuff needs to be in the water at high temperature and for a certain amount of time to heat up or it's like you didn't heat it at all. It's not the best option as weight saver, or the cheapest IMHO and with those you better stick to the cooking instructions.
Also consider couscous, it cooks really fast, or thin rice noodles.
For the instant mash potatoes in powder those can be made with cold water, you have to get the worst kind that have only powder with no flakes. The powdered stuff is not that great so you might want to give it a try at home (on a trail all the food is ok even if disappointing at home) and maybe add some milk powder, gravy powder and butter powder (if you can find it) and some spices to turn them from barely edible to barely passable.
I don't know your tastes in food but going with all the normal grocery store instant foods the more spices you have at hand the better because they tend to be very bland.
Also remember to keep an eye on proteins, as you will have to add those too and keep in mind that instant food has less nutrition than what you would have cooking it yourself from raw.
Or you can go the MRE way, those with the self heating chemical pack, the only water needed in those is the one for the chemical reaction, you don't drink it, don't keep it and it's not in direct contact with food so can be any water you find. They are not cheap if you don't have a "direct source" and for a whole week it will be sad but covering 2 or 3 days with those might mean the weight savings you want. Or FSR for extremes, but only if you plan to be too tired to be thinking about how sad is the meal you are eating ;)
With a diet of instant rice and instant potatoes (and the military stuff) for a week you better bring some form of fiber supplements or you will cut down on bathroom breaks quite a bit ;).