Have you starting to think about Christmas gifts?
Yes, it's either "have you started to think" or "have you started thinking". Both mean the same thing. I don't know the name of the tense but it's used when the action was started in the past and is still ongoing. The word order here is for a question about the action, of course.
Edit: You could also say "Are you starting to think". That is present continous tense:, i believe. But the example as you gave it is not grammatically correct for any tense.
Edit #2: The phrase "have you started" (question) or "you have started" (statement), uses the "present perfect" tense. Its use is discussed here: https://www.englishclub.com/grammar/verb-tenses_present-perfect.htm. It can be used to talk about a continuing situation that was started in the past, but not only that. In this particular example, "started" really serves to indicate that this is the sense of this tense here. This answer might be coincidental to your original question that was sparked by seeing an unusual, unrecognized construction that turns out to be non-grammatical.
Hi, I would never say 'have you starting ..'
Better to say.. "have you started to ....," or "are you starting to ...."