I am pretty sure that the rules never clarify beyond what you have already quoted, and I am certain that the rules you have quoted are not definitive on this point.

I, and literally everyone I have ever played with or talked to, have always assumed that casting spells with both material and somatic components takes one hand. Never really paid very much attention to it. But I suppose the assumption was either that the material component is consumed before the gesturing takes place, or else that the gesturing is done with the hand holding the material component.

Your GM is correct. Spellstrike doesn't give you a free cast, what it allows you to do is turn your melee touch attack from casting a spell into a normal attack, allowing you to deal your weapon damage in addition to the spell.

At 2nd level, whenever a magus casts a spell with a range of “touch” from the magus spell list, he can deliver the spell through any weapon he is wielding as part of a melee attack. Instead of the free melee touch attack normally allowed to deliver the spell, a magus can make one free melee attack with his weapon (at his highest base attack bonus) as part of casting this spell.

Spellstrike Description on d20pfsrd.

So assuming I've got all my rules straight (someone please double check me on this) you would get a total of 5 melee attacks (4, 4, -1, 4, 4) plus one spell touch affect if at least one of those melee attacks hit. You use spell combat to cast a melee touch attack spell, which spellstrike turns into melee attack at highest BAB. You get your normal attacks, and your attacks from Speed and Sudden Attack. Since missing doesn't cost you your touch attack charge, you can keep trying to deliver your spell with each attack.

As Louis Huppenbauer pointed out, both using the Arcane Pool and Sudden Attack are swift actions, so making 5 melee attacks relies upon you having applied Speed in a previous round (it has a duration of 1 minute).