I would not use "specially" in the examples you had. 1) She is specially beautiful - I would say, "She is especially beautiful." 2) It is raining specially a lot -- I would not say this, and the meaning of this is unclear to me. And how about the following? 3) I specially like him -- I especially like him for the job, because he has skills others don't (meaning I like him over the others) 4) I specially hate him -- I especially hate my boss when he yells at all the workers. (meaning : I dislike him the most when he does that) I think you would be safe in using "especially" for all those you wrote and using "specially" ONLY to mean "for a particular purpose" The meals were specially prepared to showcase the local seafood. (=prepared with the purpose of showcasing the local seafood) The car was specially designed to meet all the latest safety guidelines.

I would say that the "specially" is used 90% of the time with the verbs you listed ("make", "prepare", "give"). To me at least, the four different sentences you wrote sound kind of unnatural. "I especially hate/like him" sounds better to me. The meaning would mean that you like/hate that person in particular compared to other people. If you wanted to convey that you like a person in a special way, the only way I can think to say that at the moment is "I like/hate him in a special way." Let me know if you have any questions!

The words especially and specially have just a hair's breadth of difference between them. Both can be used to mean "particularly." Especially tends to be more formal, while specially tends to be more informal: ...