Short Answer

As long as the item was gained within the rules of AL, there is no rule that supports a DM disallowing use of resources gained between sessions.

Long Answer

Suspension of disbelief is vitally important to organized play; it is almost a core precept. Characters are highly likely to run across other characters who have played the same content, but in a different order and with different players. Each player must disregard these discontinuities, because they are inevitable unless you never roam tables¹.

The DDAL FAQ (in the DDAL Player's Pack) provides explicit guidance under Leveling Between Sessions. The short version is: That's when you're supposed to do it - between sessions. In fact, this where the "suspension of disbelief" is explicitly mentioned. It doesn't just call out the players needing to do it, it calls out the DM need to do it, too. It also talks about new equipment, new magic items, and new class features.

When you're continuing a multi-session adventure, it instructs you to keep track of expended hit points, spells slots, and other consumables that refresh on a rest. If you're down 50 hit points and have used three first level spell slots left when you leave the multi-session adventure, you're down the same 50 HP and the same spent spell slots when you come back to it. You may have a higher HP total and additional spell slots because you leveled up, of course.

Specific Examples

#1

While there are story ties between the three modules in each trilogy, they are different modules as far as DDAL is concerned. They're one-and-done. They may make more sense if they're played sequentially and continuously, but there is no expectation that they will be. They should be treated like any other single modules.

#2

By the same token, each session stands alone. DDAL expects that characters will be used between sessions of the same hardcover. See "However..." below; the GM may be making an unreasonable request. In any case, he does not have the authority to deny you the use of properly earned and properly logged XP, GP, DT, Renown, or magic items. If you're following the DDAL rules, he should be, too. Those rules don't allow him to deny it, unless he's accusing you of cheating.

#3

Only example situation #3 is a "multi-session" adventure, and the passages referenced above cover that. Note that each "part" of the introductory adventures should be logged as an entirely separate adventure - because they are. They follow the same principles as I mentioned in #1 above.

However...

A DM could ask you not to play the character between sessions², but it's not anything the DM can demand under DDAL rules. It's just a request between two (presumably) friends. Whether or not this is a reasonable request varies by the situation - how long it's going to be between sessions, etc. What is acceptable in that group is a social contract question far beyond the scope of the original question here.

The DM always has the right to tell any player to leave the table. If the aforementioned request was polite and reasonable, and the player disregarded the request anyway, that makes the player the jerk. The DM is totally within his rights as a DDAL DM and a DM in general to give the disrespectful player the boot.

¹_{If you're never going to play the character at another table, not only is the original question moot, but why bother playing under DDAL rules anyway?}

²_{I always ask this of my weekly Hardcover players. Only one has ever disregarded the request, but he was at a convention and didn't have another T2 character to play in the content and schedule that was available to him, so I let it slide.}