As far as a bat can communicate with another bat, yes.
The spell states:
Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.
That would imply that the familiar looks and acts like the chosen animal, but is not a beast when it comes to spells and skill checks. After that it's up to the DM to decide if a regular bat of 2 Intelligence can comprehend concepts of time and people well enough to tell you whether they've seen anything interesting. I doubt a bat can tell another bat that they've seen a group of people come through there yesterday.
That would still mean that "Speak with Animals" is better because it's a magical way to comprehend and communicate with animals and at the very minimum it will enable them to tell you about nearby locations, monsters and recent events they've witnessed. It seems to me that "Speak with Animals" enables the beast to communicate something it's retained far better than it could even to a member of it's own species.
No.
Consider the following excerpt from the AL FAQ:
What Rules Do I Use
As a D&D Adventurers League Dungeon Master, you are empowered to adjudicate the rules as presented by the official materials (PHB, DMG, MM, etc.). Run the game according to those rules, but you are the final arbiter of any ambiguities that might arise in doing so. (Emphasis mine.)
We see this also in the AL DMG which says in "Playing the Dungeon Master"
You're Empowered. Make decisions about how the group interacts with the adventure; adjusting or improvising is encouraged, so long as you maintain the adventure’s spirit. This doesn’t allow you to implement house rules or change those of the Adventurers League, however; they should be consistent in this regard. (Emphasis mine.)
DMs are simply not allowed to change rules, houserule, or intentionally mis-read rules.
But DaleM first made an excellent point:
The DM is allowed to be wrong!
Most of your examples look like places where your DM has made a mistake. I'd submit, though, that at least in bullets 2 & 4 those are reasonable mistakes to make. In (2) during combat they may not have the complete text of command at the tip of their brain. Likewise for (4) the rules on how adventures continue over sessions aren't referenced very often. We simply don't look over those rules very frequently, so they're not at the tip of the brain. Did they handle it well? Maybe not. Is that understandable? Maybe.
Not only are they allowed to be wrong, I'd expect my AL DM to make mistakes some times. The GM has a lot of chances to make a mistake during a session. We're not going to get every one of them right. I, personally, often lean on my players (half of whom are excellent GMs, too) to keep me on the "straight-and-narrow."
So it comes down to this: what happens when my AL DM is wrong?
You've said "when I told him about the existing rules, he either discarded what I had to say or decided to rule otherwise anyway." I think we're missing an important bit of context here: was that during the middle of an encounter? At the end of a session? Before the beginning of a session? Offline?
There aren't many players whose definition of "fun" is gathering at their FLGS to spectate as two people debate rules while their characters' lives are on pause. I'd suggest that in real time isn't the time to press a disagreement. Raise it once? Sure. But if they don't immediately see your point then jot yourself a note and move on.
Raising rules-questions out-of-session.
Ask your DM if you can chat with them while you help clean up minis and fold up maps. Explain exactly what you said above: you love some things about their style (and are trying to learn from them) but other things make it hard for you. Your understanding of AL is that they're supposed to hew to the rules. You understand they'll make mistakes just like any human, that they have many more opportunities to make mistakes, and their mistakes are more visible/impactful than any others. And ask how you can help.
If that gets nothing, if they show no interest in reducing their error rate, that's when you ask the site coordinator for help.
Best Answer
It is Adventurers League-eligible for Season 10 and legacy characters.
This September 2020 tweet by the D&D Adventurers League account states that the Celebration legacy awards are eligible for use with Season 10 and Legacy Characters:
For other games, ask your DM.
For other games, obviously, ask your DM. The oblex spawn is significantly more powerful than all of the usual options for find familiar. It is not normally a valid familiar option, as this certificate is not official D&D 5e rules content.