Question about tense. Ty for helping.
Q1: There's a swimming pool near my house. I go ____ every day.

A: swimming ( correct)
-to swim (why this is wrong?) simple present tense

Q2: Jon wants to drive up the coast to San Francisco, but I'd prefer ______(go) south to San Diego.

A: to go (correct)
-going (why this is wrong?)

Q3: For the last six months Carolvn _______(be) in Mexico－at the moment ______(work) in a school as a teaching assistant. She's coming back to England next month, and when she comes back

A:
has been ( perfect tense)(correct)
she's working. (present continue) (correct)

Is it "For the last six months"(Recently) different " last six months"(Past)?
at the moment(now?)

So, "has been" and "she's working" are correct?

was ( past tense) [last six months]
she works ( simple present tense) [job]

Why these are wrong?
We should describe a job with the present tense, don't we?

Question

Question about tense. Ty for helping.
Q1: There's a swimming pool near my house. I go ____ every day.

A: swimming ( correct)
-to swim (why this is wrong?) simple present tense

Q2: Jon wants to drive up the coast to San Francisco, but I'd prefer ______(go) south to San Diego.

A: to go (correct)
-going (why this is wrong?)

Q3: For the last six months Carolvn _______(be) in Mexico－at the moment ______(work) in a school as a teaching assistant. She's coming back to England next month, and when she comes back

A: 
has been ( perfect tense)(correct)
she's working. (present continue) (correct)

Is it "For the last six months"(Recently)  different " last six months"(Past)?
at the moment(now?)

So, "has been" and "she's working" are correct?

was ( past tense) [last six months]
she works ( simple present tense) [job]

Why these are wrong?
We should describe a job with the present tense, don't we?

Answer

1) I go swimming in present continuous. You have been going and you are still going swimming. You could say "I go THERE to swim every day" though.

2) "going" would only be correct if it said "but I prefer" in the present tense. But, it says "but I'd (I would) prefer", using the conditional, so it is "to go".

3) This is a tricky one. When I first read it I thought "was" was correct as well. The giveaway is the "- at the moment" part. It means it's still happening, so you need "has been". She has been there and still is ("at the moment"). You can't use "she's working" because you're talking about the PAST six months (and now).

You can describe a job in a lot of tenses:
- she worked there (Past. Doesn't work there any more)
- she has been working there (she has worked there for a while and still works there)
- she is working there (present)
- she will work there (sometime in the future)

Answer

1. The verb in the simple present is 'go'. To go swimming is a collocation. You use go+swimming together. You only conjugate the 'go' part.

2. Again, the verb you are conjugating is 'prefer'. We say use the infinitive with prefer. So, I prefer to eat alone, I prefer to watch tennis, she prefers to make pancakes. In your example you have used prefer with the modal 'would'.

3. She has been working shows that she started in the past and her actions continue to this day. She was working shows that it was over a period of time but now it is finished.