Why do the lanthanides sit in a separate strip below the main table?
It is a layout convention: the 15 lanthanides and 15 actinides belong to periods 6 and 7, but keeping them in the main grid would require widening the table to 32 columns instead of 18. That is why these two series appear in their own rows below the main table, still tied to the period they actually occupy.
Do elements in the same group always share the same number of valence electrons?
In the main groups, 1, 2 and 13 through 18, yes, and that is why they react in similar ways, like the 6 alkali metals of group 1. In groups 3 through 12, the transition metals, filling of the d orbital breaks that regularity, so a direct valence-electron count per group does not hold the same way.
Why does electronegativity rise across a period but fall down a group?
Across a period, nuclear charge grows without adding a shell, so the nucleus pulls the valence electrons in more tightly and electronegativity rises, peaking at fluorine, near 4.0 on the Pauling scale. Going down a group, each new period adds a whole shell, which moves the valence electrons farther from the nucleus and weakens that pull.