Learn English – Does it hurt? Vs. Is it hurt

What does "I don't think so especially that she can really play great tennis after a year break" mean?

This is a transcript of an interview. Note that "I don't think so" is an answer to the question, "So for you nothing changes when she comes back?'

How were things before? Agnieszka Radwanska, who is giving the interview, and Maria Sharapova have been competitors, both top-ranked. Agnieszka is now ranked sixth among women; Sharapova has no formal ranking because she hasn't played for a year. So "I don't think so" means that nothing changes in terms of MS being a competitor to AR.

Nothing changes, "especially that she can play really play great tennis after a year break."

"especially that"--she might have said this better, but what could she mean? Especially since, especially because, especially in view of the fact that? AR recounts that MS can play great tennis after a year break. She is referring to a shoulder injury in which MS, in the top five, was out for a year and returned to come back, after a few defeats, to the top five.

AR goes on to say only positive things about MS--she won't lose match rhythm, she will be a dangerous opponent from the first round.

You can play her in the first round, it’s going to be an interesting draw with her not being seeded. But yes, for sure it’s not going to be a good draw.

I am not an expert on seeds and the draw in tennis. However, the seeded players--the top 32--are set up so they don't play each other in the first round. The draw is (apparently) both the chart of who plays whom at each stage and the process of randomly drawing from successive seeds and eventually from unseeded players and placing them on the chart. If MS were seeded, AR would not have to play her in the first round. But in the present circumstances, a top-seeded player could end up playing her in the first round--and "for sure it's not going to be a good draw." This is further confirmation that AR views MS as a formidable opponent.

So again context matters--especially the question being answered, as well as the tennis histories of the two women and what AR is talking about when she speaks of "the draw." The OP is right.

Does he have it? 
Do they have it? 

In the sentences above, "have" doesn't agree with anything.  It's a bare infinitive form, not directly attached to any subject.  The verb that agrees with the subjects in these questions is to do. 

These are questions.  The first word of the verb construction appears before the subject.  Despite the separation provided by subject/auxiliary inversion, the verb constructions here are "does have" and "do have". 

There is no "does has" construction in English, since both "does" and "has" are finite forms.  They don't naturally combine.