Question 1

The Google translation is correct. Keep in mind that you can’t translate word for word a language.
so “I have improved in math” would be “Je me suis amélioré en maths”

Question 2

Google isn’t correct, you’re right. The correct translation is “Je n’ai pas que amélioré mes compétences”.

Hope I helped! Use this website for translating (I’m french, but I use this for German): http://www.systranet.com/translate

The first sentence needs more context. Does the sentence end there? Or are you trying to say “I have improved…” followed by a noun?

If all you’re saying is, “I have improved.” in the sense, “I have gotten better,” then

“Je me suis amélioré(e).”

is perfectly fine.

The verb “s’améliorer” is reflexive, which is why Google Translate used “être” for the auxiliary in the past tense.

Otherwise, “to improve (something),” or “améliorer (quelque chose)” in French, follows transitive verb rules in French, which means you would use “avoir” as your auxiliary in the past tense.

So if you just use “avoir” in place of “être” in the second sentence that Google Translate provided, it would work fine:

“Je n’ai pas seulement amélioré mes compétences…”

If you’re looking for sites that give you more examples and have their own forums, I use WordReference.

Q1: Je me suis amélioré is technically correct but I would prefer J’ai fait des progrès which is more idiomatic.

Q2: I would say Je n’ai pas amélioré que mes compétences although compétences might not be the best translation. We do not know what else has been improved. Perhaps Je n’ai pas fait de progrès qu’en technique would better translate that sentence.