I think the answer is yes, they can (to some degree).
The dementors are not portrayed as mindless animals, they can be negotiated with or ordered around, not merely compelled or driven. Dumbledore talked to them during the search for Black, The ministry orders them around (and I can't imagine Fudge being able to cast a patronus to compel them), Voldie negotiates with them, and they are swayed by his promises.
If dementors could not either lessen or direct their dread (to some degree), I can't imagine how they might have been allowed in a courtroom, or a school (even to deal with supposed criminals), or be sent secretly through the muggle world to assasinate someone the ministry would like silenced, as Umbridge did. This is complicated by the fact they often won't - they had to be forcibly stopped from attacking the 'wrong' people more than once, from targeting Harry even before Black during PoA, or the way it is suggested they used the excuse of Black to disrupt the Quidditch match, to Umbridge needing to use her patronus in DH to keep them under control.
As for Death eaters not needing patronuses, rather than being immune, it makes more sense that there are simply other ways of dealing with them than patronuses. As seen in DH with Umbridge and PoA with the students, the dementors may choose to go after those they are supposed to leave alone if they think they can get away with it, and the Death eaters in azkaban do seem affected (so it might not be about their natures).
There have to be other ways of dealing with or controlling dementors - ministry personnel might have to deal with them yet might not be able to use a patronus well enough (which seems to take a lot of skill) continuously or at all (I can't imagine Fudge casting a patronus, yet he has no qualms ordering dementors into a school). Perhaps some kind of medallion or other artifact to lessen or stop effects, some compulsion or control spells, or something that protects or alters the emotions of an individual so dementors can't meddle enough to feed.
Harry might have not been taught those other spells because they are not for casual self-defense (compulsion would be restricted, medallions would have to be crafted beforehand, spells that meddle with emotions would be more dangerous to the user, some spells might take dark magic, etc).