This is not a precise answer, mainly some thoughts.

Historically both ideas seem very old (2000+ years), and sometimes it is hard to tell what point of view is predominant.

Say if one looks at the Method of Exhaustion or things around Zeno's paradoxes what does arise 'sequences' or 'series'? For some of the constructions, the one seems more natural for others the other. In any case, in some form series already arose then.

One more point in favor of the fact that series where around early on: while in today's courses differentiation comes before integeration, in an intuitive sense I think integration is rather the easier or more natural idea, and historically early forms of integeration are very old (cf. Method of Exhaustion, which in my opinion counts as some sort of integeration). And integration and series sort of go together.

For your second question, for a course today, I would however start with sequence.
If one wants to talk about series in a somewhat rigorous form just having the notion/word sequence at ones disposal already seems like a big plus.

This is not a precise answer, mainly some thoughts.

Historically both ideas seem very old (2000+ years), and sometimes it is hard to tell what point of view is predominant.

Say if one looks at the Method of Exhaustion or things around Zeno's paradoxes, what does arise, 'sequences' or 'series'? For some of the constructions, one seems more natural, for others the other. In any case, in some form of series already arose then.

One more point in favor of the fact that series were around early on: while in today's courses differentiation comes before integration, in an intuitive sense I think integration is rather the easier or more natural idea, and historically early forms of integration are very old (cf. Method of Exhaustion, which in my opinion counts as some sort of integration). And integration and series sort of go together.

For your second question, for a course today, I would however start with sequence.
If one wants to talk about series in a somewhat rigorous form just having the notion/word sequence at ones disposal already seems like a big plus.