If a Devil Fruit user has a child with a person or another Devil Fruit user, would said child have a similar power, the same power, or no power and just be a regular child? Because from what I've seen in a lot of superhero movies, when they have kids, their kids might have the same or similar power. Would Devil Fruit powers be any different from that?
Short answer: No, this has not been shown, and there is overwhelming evidence to suggest the opposite.
In the world of One Piece, it is well established that any power gained through eating a devil fruit will be unique in the world. If there were a bunch of kids running around who had inherited the same powers as their parents, we would likely have seen someone like that by now or heard mention of it. If anyone is shown with some sort of a power, the first reaction of the other characters is always "He must have eaten some sort of devil fruit" and never any suggestion that the power could have been gained in another way.
It would also complicate the issue that once the devil fruit user dies, the devil fruit spawns again elsewhere in the world with another fruit, so if this were possible, you could potentially have a family where all the members are inheriting a power from some ancestor, and once he died someone else ate the fruit and passed the power to his children, and by the present hundreds of people could have the one power.
As of chapter 824, we have never to my knowledge seen a confirmed devil fruit user who has a child, and there are a couple of possibilities for this. One is just chance, the majority of parents we see in One Piece end up dying in flashbacks, and devil fruits are reasonably rare, and even if we were to believe, for example, that Edward Weevil is Whitebeard's son, there is no sign that he has the powers of the tremor fruit. It is of course possible that he was born after his father ate the fruit, so this is not necessarily a counter argument.
The other interesting possibility is that eating a devil fruit might leave the user sterile, as I said, we have no confirmed cases of an ability user having a child after they gained their powers, and in many characters case such as Luffy, Magellan, and all Logias and Zoans, there must have been huge changes to their bodies as a result of them having eaten the devil fruit.
Two more possiblities are that eating a devil fruit has no effect on your children, or that it significantly lowers your fertility without reducing it to zero. If Superman had a son, you would expect that man to have super powers, as he is genetically an alien, however if someone like Spiderman, who was born a normal human but later gained his powers through an event that potentially didn't alter his DNA had a child, you would expect that child to be biologically a normal human. If it is that eating a devil fruit lowers your fertility massively, then it is also possible that eventually a devil fruit user will successfully have a child who will inherit their power, and potentially lead to a dynasty of ability users.
Unfortunately no. When a devil fruit user dies, the fruit will respawn two years later at a random place (not really, the devil fruit power will transfer to the nearest fruit available and the ordinary fruit will become a devil fruit. We really don't know how long it takes to respawn.) The only exception that I can think of but not confirmed is that Blackbeard can transfer devil fruit from people to people. I don't know how he does it but I have two speculations. One is that his darkness darkness fruit can pull out a devil fruit from a person (most likely who is close or is dead like how Whitebeard was) and passes it to others. Otherwise he might be having a fully matured ordinary fruit with him after killing the devil fruit user. That way the powers move to the fruit since it is closer.