Under what circumstances were F1 teams allowed to re-use old chassis built by other teams?

In theory every team is supposed to design an manufacture their own car - as the '97 Concorde Agreement puts it:

A constructor is a person (including and corporate or unincorporated body) who owns the intellectual property rights to the rolling chassis it currently races and does not incorporate in such chassis any part designed or manufactured by any other constructor of Formula One racing cars except for standard items of safety equipment. Provided always that nothing in this Schedule 3 shall prevent the use of an engine or gear box manufactured by a person other than the constructor of the chassis.

Simple right? No using another team's cars. Except this is F1 and it's never that simple.

The rules as to exactly what can and cannot be shared between different teams and how different a car has to be from another to be considered a team's own have varied quite a lot over the years (see Racing Point getting caught out with the change in legality regarding sharing brake ducts from '19 - '20), and there almost always exists the get-out-of-jail route of getting the FIA and the other teams to agree to something.

So to look at your specific scenarios:

In 1992, Andrea Moda were not allowed to use an updated version of the Coloni C4 chassis from 1991.

The problem wasn't with the use of the Coloni chassis itself - but rather that Andrea Sassetti had bought the Coloni cars and the team but not the entity that designed/manufactured the cars (the "constructor") and therefore deemed them a new entry to the championship. Sassetti didn't see it that way and they hadn't paid the entry fee as a new team, after all he'd run the team in the closing races of the '91 season (still under the Coloni name) without issue. But the governing body disagreed - so officially the team wasn't allowed to enter the race in South Africa because they (Andrea Moda) weren't in the championship.

Had they ponied up the 100k and entered on the spot they wouldn't have been allowed to use the C4B as the car wasn't substantially different from the C4 that Coloni had run the previous year (IIRC there were some rear suspension changes but nothing huge).

The C4B was only ever intended as a stopgap while work on the S921 was completed which is why when they actually managed to get their sh#t together (for certain loose definitions of the term) for the third race in Brazil they used that.

Arguably if the team hadn't been such an obvious omnishambles (I think they turned up on SA with only one functional car and turned up in Mexico without any cars at all) they might have gotten dispensation to run.

In 2006, Toro Rosso were allowed to use an updated version of the Red Bull RBR1 chassis from 2005.

There was a fair amount of grumbling about this at the time IIRC, because the team formerly known as Minardi had permission to run with restricted V10s in '06 and Red Bull had the rights kicking around to this V10-suitable chassis (the RBR1 which was actually just a rebadged Jaguar Racing chassis).

The flagship team wasn't going to be using anything to do with the RBR1 since the RB3 was a fresh Newey design, not an evolution of the previous year's car. Scuderia Toro Rosso weren't a "new" entrant (as it was a continuation of the Minardi team's entry) and argued that IP rights to the RBR1 chassis came to them by way of Ford (the parent company of Jaguar Racing), and with Ford no longer being an F1 "constructor" they weren't using another constructor's chassis.

That same year, Super Aguri were not allowed to use an updated version of the BAR 007 from 2005, but they were allowed to use an updated version of the Arrows A23 from 2002.

In contrast to the STR scenario above BAR (or BAR-Honda as they were known at the time) were still a constructor, so Super Aguri couldn't use one of their chassis. Arrows were long gone and the IP and chassis had been bought from Paul Stoddart (who himself was no longer a constructor).

Of course in '07 STR went on to use the STR2 (a modified RB3) which they got away with because the rights were owned by Red Bull Advanced Technologies not Red Bull Racing and they used this loophole until it was closed in 2010 (via the 2008 Concorde Agreement). Super Aguri did similar in '07 and '08 having purchased the IP for the previous year's Honda cars via a third party.