My character was raised by elves and never learned Common language. Throughout his adventures, he had learned some words of Common, here and there, mainly daily words, like "name", greetings, until at least he (I think) should be able to understand conversations and respond well, except complex topic.

My previous AL DM allows me to understand quest briefing, conversation with NPC, without me having the proficiency in Common language. However, as I'm going to try different DMs, I'm unsure that such thing should be allowed: understand a language you're not proficient with.

Is there a rule in the book that supports that PC can understand typical conversation in a language they're not proficient with?

If yes, then what is the point of having language proficiency? (But that might be better for other question)

Normally speaking, no, there are no rules that support that you can understand a language you are not proficient in. You might roll some Intelligence or Wisdom checks to figure out what's going on, but you won't get it.

However, in the specific case of Common, there is also no way within the rules to not speak the language, so it's pretty easy to rule that your PC can understand typical conversation in that language.

After all, going by pure mechanics, especially in AL, you do speak Common, because it's just a thing you get automatically.

Your DDAL character must be proficient in Common

DDAL DMs are not allowed to make or use house rules. For the purposes of character creation, your table DM does not exist: the DDAL rules describe the character creation rules that apply to all DDAL characters and make no provisions for dropping language proficiencies.

You can understand any language by using magic

If you are a Bard, Sorcerer, Warlock, Wizard, Arcane Trickster Rogue or Eldritch Knight Fighter, you can use the ritual spell comprehend languages (PHB, page 224).

For the duration, you understand the literal meaning of any spoken language that you hear. You also understand any written language that you see, but you must be touching the surface on which the words are written. It takes about 1 minute to read one page of text.

If your character can't cast comprehend languages, you could try finding a helm of comprehending languages (DMG, page 173) which lets you cast it:

While wearing this helm, you can use an action to cast the comprehend languages spell from it at will.

No, not by RAW. However, in a non-AL game, given enough time, you can gain proficiency with a language.

Xanathar's Guide To Everything has rules for this in the Downtime section, under Training. It states the following:

TRAINING

Given enough free time and the services of an instructor, a character can learn a language or pick up proficiency with a tool.
Resources. Receiving training in a language or tool typically takes at least ten workweeks, but this time is reduced by a number of workweeks equal to the character’s Intelligence modifier (an Intelligence penalty doesn’t increase the time needed). Training costs 25 GP per workweek.
Complications. [...]

You can easily modify this and allow listening in to conversations to count as a few workdays for the purpose of this "downtime" activity. That is, however, not legal in AL (since it'd be a house rule, which are not permitted). Otherwise, your DM still has to explicitly approve this modification (obviously).

Similarly, your DM could rule that (regardless of whether listening to conversations reduces the required downtime) the closer you get to the end of your language-learning-process, the more pieces of conversations you understand. Again, this is not supported by RAW, so it's not legal in AL.

On a side note, bear in mind that, according to XGtE, you should avoid too many breaks in between spending your downtime:

The days of an activity don’t need to be consecutive; you can spread them over a longer period of time than is required for the activity. But that period of time should be no more than twice as long as the required time; otherwise you should introduce extra complications (see below) and possibly double the activity’s costs to represent the inefficiency of the character’s progress.