You could say : "I have acquired a spirit of enquiry". It would be very formal, even literary. You can talk of having a "spirit of e.g. camaraderie, enterprise etc." Some "spirits" are part of compound nouns e.g. team spirit. Your sentence sounds like part of a CV or covering letter, in which case you could say this instead: "I have learnt to approach new challenges with an open and enquiring mind; I have acquired problem finding and solving skills, as well as keeping a positive attitude while finding solutions." Remember no contractions e.g. "I've" in formal writing. "as well as + verb-ing"

This isn't a good use of the word "spirit". In English it has two primary meanings. 1 - Supernatural being of some sort. Like a ghost or phantom. In this case, it is thought of as an individual thing ("a spirit", "the spirit", "3 spirits", etc.). It can also be used as an adjective to refer to supernatural things ("the spirit world"). 2 - Enthusiasm. In this case it is not thought of as an individual things, but rather as a quality ("He has a lot of spirit.", "You have very little spirit.", etc.). So as you can see, "spirit" is not really used correctly. If you're referring to a new way of thinking (referring to your new ability to solve problems and stay positive), you can say "I have a new mindset...". I'm not sure why you used the word "doubt". It doesn't seem to fit in with the rest of the sentence. Also keep in mind that it's a noun or verb, not an adjective.

Thanks！ I'll keep that in mind:) @Peachey

Just to add to the answers below - always be wary of making compound nouns, as they can often sound rather awkward to native speakers. You're right to have asked about "doubt spirit", as it does sound strange! Sometimes we need a possessive or even a proper adjective. There are no simple rules for this, sorry.