"then" vs "and then"

I'd recommend that the non-prescriptivist view given in AHDEL be accepted as most sensible here:

Usage Note: Sticklers for grammar sometimes assert that then is not a coordinating conjunction, and that the sentence [period omitted here]

She took a slice of pie, then left

is thus incorrect; it must be rewritten as

She took a slice of pie and then left

in which the then acts as an adverb and the halves of the compound predicate are linked by the coordinating conjunction and.

But this use of then as a coordinating conjunction is actually both widespread and widely accepted; in our 2012 survey, more than three quarters of the Usage Panel found the sentence She took a slice of pie, then left completely acceptable. Note, though, that the punctuation of this conjunctive use of then differs from the usual punctuation for similar sentences using and. No comma is needed when and links the parts of a compound predicate, as in She took a slice of pie and left. When then joins the halves of the predicate, a comma is usually required; only 8 percent of the Panel approved of She took a slice of pie then left.

Though the repeated subject is deleted in this example, the accepted role of then as a coordinator is spelled out. She took a slice of pie, then she left is not too far from the example sentence.

You are attempting to join two independent clauses with a comma. This is called a comma-splice error.

Here are your customary choices:

1. I need to do something; then I’ll call you.
2. I need to do something. Then I’ll call you.
3. I need to do something, but then I’ll call you.
4. I need to do something, and then I’ll call you.
5. I need to do something and then I’ll call you.
6. Disable Microsoft.

I of course suggest the lattermost option. :)

This is from Cambridge.org. They're relatively authoritative.

Then meaning next: we can use then to mean next:

For Example:

He opened the door, then the lights came on and everybody shouted, ‘Happy Birthday’.

Heat some olive oil in a pan, then add some chopped garlic and some salt.

I am a teacher of English and have experienced this kind of use confusing. At first, my learners kept using it in speech and I always advised them to avoid using and+then as it sounded incorrect in my ears.

Later on, I realised they also used it in every kind of writing assigned to them. In a form of reaction, I chose to have a quick review about types of sentences with their syntax and semantics. I hoped that it was going to add a particular value to their understanding. Surprisingly, they have not changed to using one at a time. I am afraid I misled them, but still believe the two can't go together except when 'then' refers to time. Example: I will be visiting you every now and then.