You can use either, but they mean slightly different things:
With a negative clause, "and" means you are applying the negative to both things together as a single unit, while "or" means you're talking about each thing individually and applying the negative to each one. So:
I don't have a white shirt and black trousers.
This means that you don't have ...
The sentence is not grammatically correct.
As @stevekeiretsu says in his comment, perhaps the writer meant, "I was preoccupied with getting cute girls to like me" or something similar.
In any case, "to" is not serving as either a preposition or a conjunction in this sentence. It is used to create an infinitive, "to like".
I don't think "to" is ever used ...
The answer depends on the context.
Assume you love coffee very much, the only reason you went to Brazil is coffee. You could say
When I had gotten used to coffee, I went on a trip to Brazil to try many different types.
To emphasize the fact that you're so fascinated with coffee that you were impatient to wait more, You could say
No you can't.
You need to drop the and.
The sentence then contrasts the use of not only with the natural follow up but..also.
You will find numerous examples and explanations of the usage at sources such as these.
Until and unless can be used in many of the same places, but they have slightly different meanings.
Until refers to the time that something happens, and may be used to imply a chain of cause-and effect.
I won't make any money until I get a job. ➔ I will make some money when I get a job.
She didn't return until last week.
I'm not going to ...