The expression is usually "fix" + indirect object + direct object, which is why "fix myself a cup of tea" sounds OK, but "fix a cup of tea" doesn't. In English, as in Russian, "fix" followed by just a direct object generally means to repair. It's also primarily an American expression, which explains why you didn't find many examples of fixing anyone a cup of tea on the internet. Americans use the expression, but don't make many cups of tea; Brits drink copious amounts of tea, but don't generally "fix" them for each other. Try "Fix me a drink, honey" and you'll get plenty of results. 1950s movie, American businessman returning from work.

'Make' is the most common choice by far, although 'fix' is not wrong. I'll fix myself a cup of tea... It sounds better to me that way than with 'myself' at the end of the sentence for some reason.

you will hear it very occasionally in the U.K but use "make" it is much more natural and standard.

I use both to be honest. I might depend on the dialect of english. In England I think it's "i will fix myself a cup of tea" and in the states it's "I will make myself a cup of tea".

Yes, saying "I fixed a cup of tea for myself." is correct, however, saying "I made a cup of tea for myself." sounds a bit more natural, and is used much more often. In my opinion, you can use whichever word you prefer. English speakers will still understand what you are saying. Hope this helps!