It seems to be common and accepted practice that you cannot nest tickets on the same airline.

So you cannot fly AAA-BBB on ticket 1, then a BBB-AAA-BBB round trip on ticket 2, then the return leg BBB-AAA of ticket 1. (Rather the airline insists on AAA-BBB-AAA followed by AAA-BBB-AAA.)

If it matters, AAA=EU and BBB=USA.

But are these near alternatives OK?

i) Ticket 2 on a different airline to ticket 1? If both are *A airlines? What about miles? Best to credit ticket 1 to one FFP and ticket 2 to a different FFP?

ii) Ticket 2 is BBB-CCC-BBB where CCC is same country as AAA. So not strictly nested but close.

iii) Ticket 1 is revenue (AAA-BBB-AAA) but ticket 2 is award (BBB-AAA-BBB)? Same airlines? Different (*A) airlines?

What do you mean by common practice? I constantly have nested international tickets, often with the same airline and never had a problem. I also often have 3 rental cars at the same time or 2 hotel rooms in one night. There can be good reasons for that.
Maybe others here have made bad experiences but I never had an issue.

I don't agree.... My understanding is that nesting is only taboo if used to circumvent fare conditions...

I nest US-Australia returns all the time...

Not common and accepted at all. What was your experience that makes you say that it is ?

Right. The issue comes up if you're circumventing fare rules like minimum stay or Saturday night stay. Typically the business traveler who has to be in the same city, say Monday-Friday of two consecutive weeks, would have the most to gain from nesting. I have still never heard of a case when the airline has enforced this, but it is against the rules to nest tickets in a way that violates the fare conditions.

By definition, if the two itineraries are on separate airlines, then you are not violating any fare rules. It might be overkill to also credit to two different frequent flier programs, but that would provide total safety.

Though this is completely irrelevant to the OP's question, I have to ask: why would you have multiple rental cars at the same time? I guess if you got a weekly rate, but needed to take a day-trip to a location (e.g., outside the country) that was not allowed by that rate, then it might make sense to rent a second car. Do car rental companies prevent you from having more than one car out at the same time? (Assuming that you are the only driver on the reservations.)

Yes, I mean the case where nesting is used to get cheaper tickets - e.g. outer and inner tickets both including a Sat night, but the sequential tickets wouldn't.

Example of two Mon-Fri weeks in same city would be a good one.