Honestly, I'm more familiar with the political and diplomatic side of WWII than I am with the military, so I'm going to try and respond to these questions as best as I can.
But Franco wouldn't join the Axis. See: Civil War, Spanish; Francisco, Franco.
There were three main reasons Franco didn't join the Axis:
a) Hitler sent Canaris to Spain as an envoy to convince Franco to join the Axis. However, Canaris deliberately sabotaged Hitler's efforts by telling Franco that an alliance with Hitler would be a bad idea, since he was secretly a member of the German resistance. Send someone like Ribbentrop to do business with Franco and he might leave with a very different attitude towards an alliance with the Germans.
b) If Franco joined the Axis, he wanted Hitler to guarantee Spanish control of Gibraltar, Portugal and French Morocco. While Hitler could guarantee Spanish control of Gibraltar and Portugal, Hitler didn't want to let Franco have control of French Morocco, for fear of offending Vichy. Either have Hitler decide that an alliance with Spain is worth offending Vichy or have Hitler offer something to Franco in exchange for him putting off his Moroccan aspirations until after Britain has been defeated.
c) Hitler waited too long to court Franco. Franco seemed most interested in joining the Axis either shortly before or directly after France fell. Hitler was so confident that the German victory in France would lead to Britain surrendering that he didn't pay any attention to Franco's interest. By the time he wised up and realized that a belligerent Spain could be worthwhile, Franco's interest had cooled, since it became apparent that Britain wasn't going to fall any time soon (Franco told Hitler in a letter that he wasn't going to join the Axis until he was absolutely sure Britain was on the verge of collapse). It would've been much easier for Hitler to get Franco to join the Axis directly after the fall of France, since prior to Britain's success in repelling the Luftwaffe during the Battle of Britain, most people thought a British surrender was a matter of "when," not "if."
Spain/Portugal aren't France/Poland. Three words: Mountains & Shi*ty Roads. Also, again--your over-estimating the capacity of Spanish logistics. Like I said, see: Civil War, Spanish. Not to sound overtly the pedant, but lets remember that its not a board game or an RTS--its one thing to have the forces, another thing entirely to move, equip and use them effectivly.
Blame the Germans for over-estimating Spanish logistics, not me. The strategy I'm suggesting is based off of actual German plans for the Iberian Peninsula.
As above: no BoB--lots of strategic flexibility for both the RAF and and the RN; No life or death island struggle means opportunity to switch priorities
The fact was that Hitler was never going to be able to invade Britain because he didn't have the surface fleet necessary for such an undertaking. If he'd allied with Spain and then taken Gibraltar and Portugal, it would make it much easier for German U-Boats to strike British convoys from India and the Middle East. Taking Gibraltar also would've improved the Regia Marina's strategic position in the Mediterranean.
With Gibraltar captured, Malta can only be supplied from one direction, meaning that the Regia Marina and Kriegsmarine can concentrate all their forces in the eastern Mediterranean. The capital ships of the Regia Marina were newer than the Alexandria Fleet and outnumbered them two to one. After Malta is captured, the Axis can use it as a base for their fighters, providing air cover to Italian convoys and making it more difficult for the British to sink them. Better yet, control of Malta gives the Axis a secure supply line to Tripoli and North Africa.
Maybe. I dunno why Portgual would be all that important but thats *if* they capture The Rock, and why they would bother invading Portugal when it was neutral(ish) and they would cause another nation to join the Allies?
Dolfy didn't seem to mind invading neutral countries, even if it meant that they would inevitably end up joining the Allies - see Yugoslavia.
I really, really doubt the ability of the KM and Regio Marina (?) of "clos(ing) the straights" considering they didn't even come close OTL. and more U-boat ports in the strategic sense arent going to be a game-changer. Seeing as they had perfectly good French ports i.e Lorient, Cherbourg and St.Nazaire to name a few, which in themselves attributed to a greater length to the U-boat arm; which was doomed in the long run anyway thanks to improved and increasing Allied (UK.Canada and post 12.7.41 US) ASW and Escort Carriers
Honestly, even in a scenario where Hitler gets more nations to join the Axis, he's going to lose. If AH.com has taught me anything, it's that a German victory is impossible with Hitler and the Nazis in charge.