| | Re: Many grammar questions
1. Yes, grammatically it's possible - if both the first and the second verb take the gerund then the gerunds chain up : Compare ... The doctor recommended stopping smoking. / They advised continuing investigating. However, it often sounds a bit clumsy, and people would normally, I think, switch to an alternative structure for the one or other verb to avoid it : The doctor recommended that he stopped smoking./ They advised continuing to investigate.
However, in your example it's not possible - there is no alternative structure as both enjoy and practise only take the gerund. What makes it sound odd is that, here, playing is redundant. It's enough just to say He enjoys practising the guitar.
But if you take another example - eg He enjoyed practising walking the tightrope- to me it sounds quite possible.
2. Depends what you mean by "possible". Logically no, and I'd avoid it in writing - but I could see it being used frequently in spontaneous speech where, by the time the speaker reaches the end of the utterance, they've forgotten the beginning. Spoken discourse is rarely totally grammatically logical because of the "real time" factor. And in a continuing conversation, you wouldn't notice it.
3. A pro-verb. Just as a pronoun subsitutes for a noun, an auxiliary verb (or Be as main verb) functioning as operator will often substute a complete verb phrase, where that verb phrase is understandable from the co-text. Eg John can't swim but Mary can (= can swim). John didn't go to Rome last week but Mary did (= went to Rome last week).
It can't happen though in your sentence because a guitar is the object rather than the subject : The instrument used was a guitar. So, as you say, the sentence has to be It was a guitar or just A guitar. Compare the other example you give (The alarm is making that noise) and this one : Who was the last person to arrive?/David was. In each case the "answer word" (alarm/David) here is the subject.