What would we call these fallacies?

Hello Daryl, and welcome.

I think I know what you're talking about. Let's parse your post:

When someone is giving an argument in absence of opposition, I've seen them state things as if everyone in the audience agrees with them, as if to influence people in thinking that the majority of people do agree since nobody is questioning the statement.

I interpreted this as being one of those cases where people say things like "everyone knows X" as if something were obvious, but X is actually contentious, therefore not obvious.

Depending on how it is expressed specifically, it could be prejudicial language ("all good people know X") or proof surrogate ("X is an indisputable fact no one could possibly deny" - note the use of the word "indisputable", which is masquerading as evidence for X).

EDIT: See Dr Bo's answer. "Alleged Certainty" fits this example way better than whatever I name-dropped in my screed.

When someone is giving an argument in the absence of opposition, I've seen them go a step farther and explain that there is opposition, but they are liars, are immoral, or generally not to be trusted. Or just state the opposition's argument in a way that doesn't do it justice so they can dispense with it quickly. It's basically something like: I got to this audience first, I've set the narrative, and by the time the opposition attempts to make their argument, most of the people I've spoken to know they can safely reject the opposing argument without listening to it.

Two fallacies going on here - the first is poisoning the well (this is setting the narrative that the opponent is untrustworthy, stupid, otherwise immoral etc). This primes people to see the opponent in a negative light, which makes them prejudiced against said opponent's arguments - before the audience has even seen them.

The second is a strawman fallacy. The speaker "acknowledges" the existence of counterarguments, but bastardises them such that the audience doesn't see them as serious threats to the speaker's thesis.

I've seen where string of false claims will be made that go from less to more extreme, the claims will generally be an emotional minefield that make it impossible to maneuver, and then the opposition will simply accede to all but the very last and seemingly most important claim to rebut, thereby convincing the audience that the last several claims must be true, and the only legitimate arguments are about the very last claim. In reality the argument was always about winning the previous claims by default, since the last claim will be won in a future argument by not being the last claim in a new string of false claims. I'm particularly interested in this one, because as I've seen demonstrated over and over again, nobody seems to know how to deal with this situation.

Here, I picture the speaker making a string of claims - A, B, C, etc - wrapped in extremely emotional rhetoric. The audience, or opponent, accepts all the claims up to the final one - say, D - which is swiftly contested. However, A, B and C are also questionable claims, but thanks to the emotional blindsiding effect, they've gone uncontested. Thus, the speaker has smuggled dubious claims past their interlocutor via sloppy reasoning.

The closest match I can think of is hypnotic bait and switch (but the definition implies that the claims are uncontroversial and true - I believe the format can work for cases where the claims are false or at least equivocal, but are accepted as true.)

EDIT: As Dr Bo points out, this could also be considered a Gish Gallop.

To deal with it, it's probably best to take stock of every individual point the speaker makes - if necessary, ask them to repeat themselves. You'd want to make sure you're focusing on the most relevant parts of the argument.

#1 See alleged certainty 

#2 See poisoning the well 

#3 See gish gallop