In the movie John Carter did he ever drink water? Humans needed it to survive and I thought there was no water on Mars.
The fictionalized world of Mars as envisioned by Edgar Rice Burroughs was called Barsoom. It is not Mars as we know it, but a more fantastic planet populated with a variety of creatures. Barsoom is a romantic representation of a Mars at the end of its resources with various races struggling for survival.
There is plenty of water on BARSOOM. But not nearly as much as there used to be. The last river, Iss is a customary river used by Martians who wish to seek to end their very long lifespans.
The sole remaining river upon Mars, for ages green and red Martians upon their thousandth year of life would take a pilgrimage to the River Iss, they would travel down to the river to what they believed was the afterlife.
As for why John Carter might marry one of the Red Martians:
All Barsoomian races resemble Homo sapiens in most respects, except for being oviparous (egg-laying) and having lifespans in excess of 1,000 years unless killed.
The traditional Martian lifespan of 1,000 is based on the customary pilgrimage down the River Iss, which is taken by virtually all Martians by that age, or those who feel tired of their long lives and expect to find a paradise at the end of their journey. None return from this pilgrimage, because it leads to almost certain death at the hands of ferocious creatures.
While the Martian females are egg-laying, Martians have mammalian characteristics such as a navel and breasts. While they have skins of various colors, and their bodies differ in some cases from traditional humans, they are very similar to varieties of Earth humans, and there is little examination of difference. There is only one spoken language across the entire planet, but a variety of writing systems.
Given the egg-laying nature of the Martians it is unlikely without technology that Carter could successfully produce offspring. Such technology must have existed since he and Dejah Thoris have a son.
It is not quite correct to say that there is no water on Mars:
However, Mars that exists in the fictional universe of John Carter might not be the same as the real Mars and might have extra water etc.
I am guessing that when you ask about the genetic compatibility of humans and martians then you are asking about them having children? Genetically anything can 'marry' anything they like - you could marry a goat if it was legal, that does not mean you could procreate with it.
I am not sure if children came up in their marriage but being as Martian and Humans are different species they, by definition, cannot have viable offspring. If they do then either they were the same species all along (perhaps descended from a lost common ancestor but considering how close we are to apes whom we cannot breed with this is unlikely) or the Martians have a currently unknown genetic or scientific enhancement which allows this.
Barsoom (ERB's Mars) was dry, but no dryer than high desert on earth. There are plants, there is some dew at night. There is enough moisture to fill the river Iss, albeit only to a low flow channel, and it covers most of Barsoom.
Where there is plantlife that needs water, there are stores of water.
It's important to note that the movie is a moderately loose adaptation (and works in elements from at least the first 4 Barsoom novels, while covering the first primarily); several elements, like John Carter finding water in the desert, are simply not covered. Still, it is fairly faithful to the first novel, A Princess of Mars in the overall story. The movie shows surprisingly great respect for the book.
Also not explicit in the movie is that John Carter was not himself entirely human. This is fairly explicit in the novels - his ability to transfer to Mars is not his only inhuman ability. Mention is made in a later novel of having been of great age already when the civil war was fought, and the first novel is set some years after that. He has some abilities of the Therns - he might be a half-human half-human Martian.
He eats moist foods, and drinks various beverages in the novels; in the movie, we see beverages upon the table, and moist foods as well. Certain of the plants can be cut and wrung for water.
Now, as for finding water upon arrival - he doesn't. What you see in the movie is, almost exactly, what the novel tells, only extended in time a bit from what the novel implies. His capture is a scant few minutes after awaking on Barsoom - tho', yes, Carter himself, in both movie and novel, recognizes it as Mars.