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You are correct that C-E-G would be the notes for a C chord (1-3-5). Now about inversion. If the C is the lowest note (most Bass note) that would be the Root inversion. If the E (3rd note) is the lowest Bass note that is the first inversion. if the G (5th note) is the lowest Bass note that would be the second inversion. If it were a C7 chord with four notes and that 7th was the lowest note, it would be the third inversion. Now I’ve never really thought about how you know which string to play the notes on. It’s always been for me which strings will my finders reach. The different chord shapes let one play the same chords in different voicings in different places on the neck and in different inversions. If I’m playing with some one who is playing an open C chord, I might play the same C chord with an A shape on the 5th fret (root on C on 5th fret) or an E shape barre or (mini F shape) on the 8th fret to achieve the same chord with a different voicing. I’m not sure if I am addressing your question, but the fact that you are asking helps us all grow. Thanks.