Are these assumptions correct?
In general - no. But on high school level - most likely yes.
Is it ALWAYS whichever is lower in the electrochemical series will form on the corresponding electrode?
No. Several reasons for that. First - potential is a function of concentration, that is described by the Nernst equation. Thus just the presence of substance is not enough, it must have concentration high enough. It should react at a given potential when concentration equals 1M, this is so called standard state - that's why you have a table of STANDARD electrode potetials, as opposed to the FORMAL ones - observed in the real solution.
Then, some reactions are sluggish and/or depending on the electrode material they don't want to start at the standard potential and they require so called overpotential to be added.