I was told from a friend that the death of the Tribunal is what helped caused the Oblivion crisis, is this true? Did their deaths help break the barrier preventing Oblivion from invading?

Short answer: no.

The Nerevarine's freeing of the Heart was the main cause, something you did on Vivec's instruction.

There is speculation that Sotha Sil's pact with the Daedra could have expired upon his death. Whether that was true, and whether it even mattered if it were true will never be known. There were multiple more important factors at work.

How did freeing the heart cause the Oblivion Crysis? and did Vivec know it would happen?

It took down red tower, which weakened the existing barrier.

VEHK may have known it, but defeating house Dagoth mattered more

It deactivated Ur Tower I thought. The Heart being its stone. I'm sure Vivec had some notion of what could happen, but it was necisary to defeat Ur and save the world.

Red Tower*

Ur Tower refers to Adamantine, and it's stone, the zero-stone, is the moment of creation.

Since destroying the heart caused it, this led me thinking...was it really Azura that you spoke to at the beginning? Could it have been Mehrunes Dagon tricking you in to believing you were Nerevarine to fulfill the prophecy and destroying Dagoth and the heart? By destroying the heart, it helped break the barrier between Tamriel and Oblivion, the powers of the Tribunal were lost which means Vivec lost his divinity which would cause Baar Dau to crash in to Vvardenfell destroying most of it. This all sounds like something Mehrunes Dagon would want.

The destruction of Red Tower didn't weaken the barriers, but it did cause a Dragon Break. With the successive destruction of each Tower, space-time weakens.

The Convention saw the creation of Ur-Tower and each one after. Now the Towers are being destroyed one by one as well: Walk-Brass in Daggerfall; Red-Tower in Morrowind; White-Gold in Oblivion. Slowly but surely, Nirn will return to Mythic Age, like the mainspring in a clock unwinding with each click.

No, it was Azura. There is extensive lore stating it was her and even a Dev rp known as The Trial of Vivec where he gives her his shaft.

As far as I know, no it didn't.

Which suggests to me that the final game of this kalpa will be set in Summerset Isle - if the kalpa is destroyed. We could see an almost Arena-like re-institution of the other Towers.

It is worrying, btw, that the Khajiit have come under the sway of the Dominion, possibly effectively rendering their "Tower" inert.

Where is that stated?

Won't a dragon break simply cause time to change? Like while playing Oblivion we will suddenly move forwards at time to the few minutes before our death then backwards to when we were born then forwards to the age of 10 etc.?

Also, if there was a dragon break during Morrowind, won't Jills appear much earlier to heal time?

So it kinda makes sense that there was no Dragon break, no? or is it just my lore-inexperienced-brain?

As much as I like this, it's simply not true.

Nu-Mantia never mentions any Towers but Adamantia, Red Tower and White Gold as being important to Tamriel's security against Oblivion. The fall of Red Tower alone is considered important in this document, opening the way for a further threat regarding White Gold.

Of course, the general forum consensus has been that all Towers fortify reality and hence form a barrier with Oblivion. New Skyrim books point to that version of things, possibly, ironically, based on popular understandings of MK's Nu-Mantia text which has an alternate sort of Oblivion Crisis envisioned within it.

Make of it what you will, but there really is no evidence of a Dragon Break at the end of TES III. The khajiit would have noticed.