If someone asked me "Have you seen the new season of GoT?" do I respond "I didn't know it had started already!" or "I didn't know it has started already!"? Or just "it started already"?
Prescriptive English grammar, as I was taught it in 1962, requires the use of the past perfect tense following a verb in the past tense. Therefore, by this prescriptive formula, only "I didn't know it had started already!" is acceptable. However, grammar changes based on usage. Descriptive grammar, the rules that govern spoken informal English, seems to be moving away from the use of the past perfect tense, at least in the US. Word Processor applications underline "had had" in red, in the expression,"He had had enough." So, my answer to your question is, it doesn't matter which expression you use, unless your audience objects. My own preference is to place the word "already" after the auxiliary verb "had", which sounds more euphonious: "I didn't know it had already started!"
All of the phrases are acceptable. The phrases have slightly different meaning, but the end result is that all of your examples are okay, and effectively communicate the same idea.
"I didn't know it had started already!"
You weren't aware that, at some time in the past, it did start. The phrase "had started" is "past tense", so the statement technically focuses on something that happened in the past.
Even though this sentence is describing something about the past, we can safely infer something about the current situation. We can assume that the status is a later status: it is active, or completed.
"I didn't know it has started already!
You weren't aware that the television season's current status is that it has started. The word "has" throws this into "current tense" (instead of "past tense"). For this particular situation, that is perfectly okay.
At a prior point of time, the status of the season was "waiting to start". That is no longer the current status. It is not waiting to start.
Or just "it started already"?
That works perfectly fine as well. This is probably preferred, simply for being short and succinct. Unnecessarily usage of "have"/"had" can result in the grammar sounding more convoluted, and many people struggle by using such phrases incorrectly. This short phrase entirely avoids the risk of incorrectly using "had".
This is using a verb that is "past tense", to describe something that occurred in the past. Nice, short, sweet/simple.
Let's just show another example, unrelated to a television series:
If a race "did start", that means that the event did begin at some time in the past.
If a race "is started", that means that the current status of the race is that it is no longer waiting to start.
Since the event only starts once, the minor differences have no effect on the overall important meaning. So, any of these phrases ends up working out equally well.