would + infinitive (would sing) = aș/ai/ar/am/ați/ar + infinitive (aș cânta)
would have + past part. (would have sung) = aș/ai/ar/am/ați/ar + fi + p. part. (aș fi cântat)
In English you would just say "with me" not "together with me", it sounds very odd.
'Inota' is with only one 'n'. If it was with two ''N''s, then it would have been a derivative from the word 'nota' wity the prefix 'in'. That, when you have a word starting with 'n' and you ad the prefix 'in', is the only time you'll see a double 'n'. So not for Înota.