Let's talk the rationale for the rules relating to the questions.

Runners need an opportunity to fix running mistakes. While the ball is live, they have every opportunity to return to a based left early, or missed. In the old days (meaning, before the dead ball appeal, and subsequent rules tweaks), once the ball became dead, they had no such opportunity. If umpires realized that runners were still attempting to run the bases, they would/should refuse to grant time.

So, what did the players do? Well, they threw the ball out of play to kill the play and any opportunity for baserunners to finish running the bases. Who cares if there is a base award to a runner that is going to be out on appeal? So, the first rule tweak was to allow a runner to continue if, and only if, they were in the act of returning when the ball became dead.

Adding the dead ball appeal made it a bigger issue. When all appeals were live ball, not everyone stupidly called time to force the ball to become live again from the pitcher's plate; but now, practically every true appeal is dead ball. What to let the runners do, what stops the defense from rigging the system? The rule and POE/RS were modified to say that runners must be given the opportunity to complete advancement; then added the opportunity to complete their base running responsibilities.

BUT, the RULE 8.7-I(3-a) only relates to legal advancement "if the ball leaves live ball territory". The RS adds that umpires should hesitate to allow completion of running responsibilities, and then (or if responsibilitiies are ignored) make base awards; because once the runner touches the next awarded base, there can be no going back. The Note: under "May Not Return" in the RS#1 is clear as the intent of all these.

So, the edited question:

No, the runner is given the opportunity to fix mistakes during live ball, or if the ball leaves the live ball territory. Umpires should not grant time when there is the live ball opportunity to fix, and absolutely not if there seems to be any inkling or recognition by the offense that may need to happen. (We grant time when all play is ended by rule, not to stop further play.) If the ball doesn't become dead by the umpire granting an unneeded "time", we haven't created the impossibility of their fix.

IMO, the base umpire should almost NEVER grant time, unless immediate danger to someone or the ridiculous hold-a-tag game has broken out; let the plate umpire with the broader over-all view knowing all play has ended by rule call time if it needs to be granted. That "someone asked for it" is rarely one of my considerations, unless I see a reason (coach wants a conference, making a sub, checking on a player, or ready to make a dead ball appeal). Seriously, when was the last time you DIDN'T know they wanted to appeal before they actually did?

From the 2013 Umpire Clinic Guide:

2. Dead. The dead ball appeal may be made:

a) Once all runners have completed their advancement and time has been called. Runners must be given ample opportunity, in the umpire’s judgment, to complete their base running responsibilities. Any infielder, with or without the ball, may make a verbal appeal on a runner missing a base or leaving a base too soon on a caught fly ball. The pitcher and the catcher are considered infielders for the appeal process. The appropriate umpire should then make a decision on the play.

I assume this is ASA only? The rules I quoted before from NFHS and NCAA would seem to assert that base runners cannot return to those bases once the umpire calls Time, and only can correct mistakes when the ball enters dead ball territory.