"The synthesist also gains access to the eidolon’s special abilities and the eidolon’s evolutions"
Does this mean that a humanoid Summoner, when fused with his quadruped Eidolon becomes a centaur with 3 pairs of legs and one pair of arms?
"Quadruped" is a base form, not an evolution, so no.
No; the summoner will need to buy the Extra Limbs(arms) evolution before he can cast spells requiring somatic components.