‘Yes’ and ‘yes, in this case’. The volitional form (it’s in the name, because ‘volition’ is about one’s will i.e. desire/intention to do something) is something that expresses the intention to do something (like ‘I shall do…’), but it can also be used for suggestions or invitations, as you’ve just seen here! 
I wouldn’t say so. と is definitely more commonly used as a quotation particle. Here’s a rough guide:
- と + verb expressing speech, writing, thought etc.: quotation particle
- と + other verb: quotation particle being used to frame/put forward a description of how that action is done (very basic and common example of this: 〜とする, as in その色（いろ）を黒（くろ）とする=‘to consider that colour as black’, literally ‘to do that colour [as] “black”’. You could even parse it as something like
‘to do/act（する）, with that action applying to ‘that colour’（その色を）, while [thinking, saying etc.], “Black.”（黒と）’ .) - noun + と (+ noun): means ‘and’ or ‘with’