What happened/ has happened here?

Both possible, very little difference in meaning here I think.

Tell me what has been happening/ has happened to you in the past few days?

I think again both are okay. They feel like slightly different questions, but it's hard to explain how, and you'd give the same answer to both of them.

She had worked / had been working / was working there for 2 years when Martin started at the company.

"She was working there for 2 years when he started" sounds more like she was in the middle of a two year contract. She joined up to work there for a duration of two years; after six months, Martin joined. The other two are both correct and both feel very similar.

He had been writing / was writing a novel for a bit when she came to talk to him.

I'm really uncertain about this one. I'd more likely to say "he'd been working on his novel" or "he'd been doing a bit of writing". But in terms of tense I think both are okay again.

The kid's hands are soiled because he had been making/ was making slime creepy-crawlies and I tried to remove the stains from his hands with liquid soap but some of them have still remained/ still remained n't come off / didn't come off.

"remained" makes sense but sounds a bit awkward. Both tenses fine still :)

One day our ancient air fryer for burgers blew out all the hot air into our colleague's face and he needed to undergo aplastic surgery because his face had been badly burnt.

The boss is mean and ctrashes everyone for the slightest mistake.

Not sure if this is what you meant. To trash someone is kind of to be really mean to them, say that they're very bad, be very disrespectful. "crash" makes no sense to me.

I'm sorry that I didn't write you an e-mail .

I just gothad so many things to do inat my job.

This "I got so many things" grammar is really common colloquially, but I don't think it's formally accepted.

I'm sorry that I haven't written you an e-mail but I've had so many things to do inat my job.

I had a load of things to do at work.

"at work" is more natural than "at my job".

Now I've found some time to scribble you an e-mail.

Which of the sentences about writing an e-mail is the most correct, or perhaps are all of them are correct?