Generally voting districts in a state are defined and controlled by the state legislature. Let's use my home state, North Carolina, as an example.
N.C. currently has 13 congressional districts. In the United States we hold a decennial census (every 10 years). There are other population surveys done in between, but once every decade is the big one where we try to count everybody. Once the counting is done, an apportionment of districts are decided for each state. The total number of apportionments is exactly equal to the number of seats in the U.S. House of Representatives (currently at 435). The Senate is ignored because its membership is exactly equal between the states, whereas the membership of the House is allocated based off of population, each state gets 1 representative per district. So in 2020 after the next census is completed, N.C. may not have 13 congressional districts anymore, it could have 12 or 14 or 15, depending on population increases/decreases relative to all of the other states.
Once N.C. receives its allocation after the census, its State Legislature is charged with coming up with new voting districts. Not all states defer to the legislature, some have a completely independent commission draw the lines. This is a good resource for more information on how each state draws their lines.
For a state legislature to adopt a new map, all that is really needed is for them to pass the new map as a state law. If no one has a problem with the new map, great, otherwise someone will most likely sue the state in federal court over it, usually on the basis of Section 2 of the Voting Rights Act which requires states not to engage in practices that discriminate on the basis of race or color.
Historically, I think the large-scale practice of gerrymandering began right before and after the Civil War. New States were being added to the union and it was a big fight every single time deciding whether slavery was going to be legal in each new territory or not. Slaveholding states wanted to protect their industry, and were afraid of new, "free" states being added because they would allow the free states to form a coalition that would outright abolish slavery in the whole country. This is similar to the reason behind there being both a North Dakota and a South Dakota rather than just a single "Dakota," because the Republicans and Democrats of the day were keen to only adopt new territories as states should it help them politically, and often times a balance of ideology was necessary in what states were admitted and when in order to strike a bargain. In the Electoral College, each new state gets, at a minimum, 3 electoral votes regardless of population, so splitting the Dakotas into two wound up doubling their influence in Presidential elections.
You should keep in mind that so far answer only addresses federal voting districts. North Carolina's State Legislature also has two chambers (a State House and a State Senate). The State Senate is composed of 50 members and the State House is composed of 120 members, so we actually have 3 different types of voting districts that we need to be concerned about: the U.S. House voting district, the N.C. Senate voting district, and the N.C. House voting district. You can see the maps for all 3 of these here. I bring this up only because the Voting Rights Act is only applicable to the U.S. House voting district map. The North Carolina General Assembly has much greater flexibility by statute to determine the boundaries of State-level districts.
From what I can tell, the answer to your last question is no. Gerrymandering as a practice is not supported by the population on either side, but because of timing of the last census with a wave of Republican takeovers of state legislatures, Republicans are generally seen as being more guilty of the practice in the current day and age. Some forget that Democrats are by no means innocent. I can't seem to find a source at the moment, but one of the arguments back in the 1960s and 1970s was that, in order to get black politicians elected, they needed a mostly-black district in order to support them. Even though the end of having a more diverse Congress can be thought of as more noble than simply aspiring for a partisan majority, the means taken wound up being the same: drawing districts such that people of a certain race were lumped together, and the same means have been turned around now in an effort to dilute the same population's voting power. Recently there has been a somewhat strong movement towards using a bipartisan, independent commission to draw maps, however a divided Supreme Court has also weakened some of the protections provided by the Voting Rights Act.
Given how ingrained in the American conscience fighting for one's beliefs are, I don't really see the practice itself dying out anytime soon, regardless of what it is called.