In the sentence, I asked him if he cooks at home. Should cooks be changed to cooked or left alone. For example if I change it to cooked it gives the sentence a different type of question, but using cooks I am asking if he cooks all the time. Does this make sense? Please help!
Yes, using "cooks" is asking if he cooks all the time or habitually.
You use the present tense if you want to know if cooking takes place on a regular basis, without any specific starting or ending date, and without being restricted entirely to the past or future.
You use the past simple tense if you want to know only if cooking took place at some time in the past (note that the answer you receive will not tell you anything about the present or future).
As you would (in most cases) suspect that it still be the case I'd say that "I asked him if he cooks at home." would be acceptable.
"If the quoted sentence expressed something thought
to be a general truth, then no tense shift in indirect speech is necessary."
page 8 of http://www.artsci.wustl.edu/~djkaiser/127/VerbTense.pdf
"If the speaker is reporting something that he or she knows is true or believes is still true, the tense of the original utterance is often preserved."
page 9 of the above.
"If the reported sentence deals with a fact or general truth, the present tense is (can be) retained. She said that the moon causes the tides.
"If the speaker reports something immediately or soon after it was said, the noun clause verb often remains as spoken.
A: What did the conductor say?
B: He said that the next stop is Northgate.
"If will is the modal in the reported utterance and expresses future time, and if the situation described in the quote still holds true at the time of the indirect report, the will may not be changed to would even though the reporting verb is in the past tense:
Mr. Arden said that a volcanic eruption will occur next year."
"Also, a sentence in direct speech in a present or future tense can remain the same if what is said is still true or relevant."
According to The Sequence of Tenses the correct sentence is: "I asked him if he cooked at home." I think the fact that somebody cooks doesn't belong to "general truth" such as the motion of cosmic bodies for example. It is much more ordinary and utilitarian.
And "the present tense can be retained", not must be. If retaining the present tense was a strict rule we would always take the risk to make a grammar mistake only because we don't know if the fact we are talking about is true.