The point is that you may be asking the 'wrong' question. The use of paths is inappropriate in this algebraic geometric context, and hence also ideas such as 'path connected' and 'fundmental group(oid)'although there are analogues.
The classical fundamental group classifies covering spaces of $X$, and that is a useful property to generalise. Look at Grothendieck's SGA1 (if you can read French) for the original material on this, but there are lots of more recent sources 'out there' (including some well written surveys, some done by various Masters students, that are a good read and get you to the point fairly quickly without a lot of generality! One such that I have used is 'M. A. D. Robalo, 2009, Galois Theory towards Dessins d’Enfants,
Master’s thesis, Instituto Superior Technico, Lisboa'. Another very good source is by Dubuc and de la Vega, (and which can be found on the ArXiv as math.CT/0009145.) There are analogues of the fundamental groupoid and of covering spaces and then you can ask if the fundamental groupoid is connected (corresponding to path connectedness in the classical topological case) That fundemantal groupoid is usually thought of as a profinite groupoid, and that may encode the topological information that you are hoping for in the second part of the question. (I should add that it is best to consider general schemes with the étale topology, but to start with that may not be necessary.)
I hope this helps.