Is it possible to rig elections?
There are plenty of criminal offences in relation to rigging elections: voter impersonation or intimidation, interfering with ballot boxes or voting machines, vote-buying, abuse of public offices. From time to time, people have been convicted of these offences. It seems reasonable to suppose that from time to time, people have got away with them. There's no undisputed example of unlawful tampering with the electoral process actually changing the result of a Presidential election, in the sense of a criminal conviction where the crime committed changed the result. You can't precisely accuse George W. Bush of rigging Florida in 2000 by winning a court case, although some do accuse him of "stealing" the election, since they don't agree with all the various courts' decisions in respect of the recount.
Some people accuse Jeb Bush of rigging Florida 2000 by purging voter rolls of innocent voters on the pretext of them being convicted felons from out-of-state, when in fact they were not. But this is hardly undisputed. He did use lists of out-of-state convicted felons to remove names. Many of the people he removed co-incidentally shared names with convicted felons. But he didn't even completely block them from voting: merely made it harder for them to vote than it was for people who didn't share one of those names. Whatever someone thinks his intent might have been in terms of disproportionately affecting Democrat voters, there were no grounds for a prosecution, so as far as the indisputables go, he was playing the game by the rules. However it was in that case near-certainly possible for a state Governor to improperly affect the outcome of a Presidential election. Whether Jeb Bush actually did so is irrelevant: the situation merely illustrates that there are mechanisms by which a Governor could act improperly and have some chance of getting away with it.
Is it possible for the American establishment and the Russian government to do the things Trump and Clinton accuse them of?
For the most part yes, it is possible for the American establisment to present a consistently biassed view against Trump, and it is possible for the Russian government to sponsor hackers to retrieve Clinton's emails or other sensitive information from wherever it might lie.
Can the things they're accused of affect the election outcome?
Depends how close it is. If it's really close then anything can affect the outcome, so it's possible. In a close count, both sides will have observers disputing the interpretation of particular ballots on a case-by-case basis, because even a (literal) handful of votes could swing the result. In a situation like that, even a small amount of electoral fraud, or a small number of people whose vote was swayed by a negative story about a candidate, could be enough.
Does the establishment of either the US or Russia get to decide the result of the election?
Not in general. Certainly they can't both have the power to just decide the result, but if either of them does then it's the US establishment in best position to somehow manipulate the voting and counting process. It's big-time conspiracy territory to suppose that there's any person who decides whether or not they like the result and flip it if not. Rather, members of the establishment seek to obstruct their opponents, and it would be astonishing if this never extends to improper or illegal interference.
It's generally considered unethical for foreign powers to interfere at all in elections, but to suppose that neither the Russian nor the US government has ever sought to affect the results of a foreign election would be naive. Both states have sponsored armed coups in other countries in the past, never mind under-the-table support for their preferred candidate. The extent to which their intelligence services get involved in campaigning in each others' elections is of course unknown, but I suppose it's safe to say that Russian elections have never been close enough that the US government has ever had any real hope of affecting the outcome. So if either of them has a chance of doing it effectively to the other one, then it's Russia to the US.
Is interfering with an election rightly described as "rigging it".
Not always. The common understanding of "election rigging", I think, is illegal interference in the electoral process itself. Illegal interference in the campaign is not usually described as "rigging", even though it's improper and it could affect the outcome at least in principle. In practice it's incredibly difficult to evaluate hypothetical questions like, "if this bad story about a candidate hadn't come out during the campaign, would they still have lost?", but pollsters can try to establish people's reasons for voting as they did, or for switching their intentions when they did.