Is the Mandelbrot set path-connected? The Mandelbrot set is known to be connected but whether it is path-connected is an open question. But what is the general consensus/belief among mathematicians? I am unable to convince myself either way because I am rather new to topology and don't yet have a good intuition about connectedness and path-connectedness.
 A: I cannot speak about the whole of the complex dynamics community of course.
If you are new to the notions of path-connectedness and local connectedness the Mandelbrot set might not be the best topic to gain intuition, since as you said those are current research areas and therefore well beyond the scope of a first course in topology.
That said, there are results (mainly due to JC. Yoccoz, a Fields medallist) proving that the Mandelbrot set is locally connected at every point except possibly some special points (called infinitely renormalizable parameters; roughly speaking they are those points that are in infinitely many nested small copies of the Mandelbrot set). That seems like a good step towards proving the conjecture.
Note that experts are not so much interested in this question by itself (local connectedness of the Mandelbrot set) but rather are interested in a conjecture called genericity of hyperbolicity, which is known to be true IF the Mandelbrot set is locally connected.
Oh and by the way I don't think there is an easy way to convince yourself one way or another (short of reading several research papers on the subject).
For the sake of completeness, here is an argument against the conjecture: it is known that the analogue of the Mandelbrot set for cubic polynomials (the connectedness locus of the Julia set) is NOT locally connected. 
