1. I won the first prize in the speech contest.

2. I won first prize in the speech contest.

(Do I have to use 'the' or not?)

3. You'll be making a business trip to Korea in an early July.

4. You'll be making a business trip to Korea in the early July.

(Do we have to use 'the ' or 'a' in #3 and #4?)

5. We are going to give/have/ throw a party.
(Are all verbs OK?)

6. Is Kelly a female name or a male name?

1 answer

2 is better than 1.

4 and 5 -- don't use either "an" or "a" or "the"
You'll be making a business trip to Korea in early July.

5 -- all the verbs are OK.

6 -- as a first name, it's mostly a girl's name, but I've seen it for both.