They're also often signed by minors who cannot be legally bound to a contract

None of these are (speaking generally) legal requirements of an agreement in the US.

Source: Am lawyer who writes EULAs for a living.

There are plenty of good arguments for why a particular EULA might be legally problematic, but "no signature, no contract!" isn't one.

It would depend on the jurisdiction obviously, but I believe most of those points are irrelevant.

• An arbitration clause is just a preference, it doesn't mean you can't resolve a disagreement in some way.
• Just because a contract typically binds both people to some requirements doesn't mean it always has to be that way. An agreement can be one-sided. In the case of EULAs though, there is the requirement on the company to provide you with the software and allow you to use it for as long as the EULA is in effect.
• I doubt an agreement has to identify both individuals in the actual text. The key aspect is whether both parties agree to the terms and whether it can be shown that the individuals agreed to them after the fact.

As far as signing goes, I know that in my country (Sweden) a verbal agreement is legally just as good as a written signature - it's just harder to prove in court. Contract law typically recognizes the ability to agree electronically, and in EULAs the agreement is made by using the software. Again, YMMV by country. My original claim that they're typically illegal was about the actual terms of the agreement, which often conflict with written law. For example in the EU you have a right to reverse engineer products for the sake of interoperability, and no EULA can override that right.

In Sweden there's also a law to allow you to make personal backups of media and software, and you're permitted to give copies to your friends and family. In fact, there's a state-regulated "private copying levy" designed to compensate content owners for their monetary loss caused by this copying. Which really infuriates me considering the lengths they go to to prevent me from doing the copying that I'm paying them for the right to do.

There's no consideration. So yeah, not a contract.

They are usually defining the specifics of whatever contract the company offers to their users, be it the sales or whatever services the company is providing. I don't know which legal system you're living under, but such additional clauses added to the base contract can very much be explicitly or implicitly agreed upon.

Edit: Of course this particular rendition of "We're allowed to break the law - no take-backsies" won't stand anywhere consumers have like any rights at all.