Would desperation allow you to attack 4 times before opponent can counter attack?

No it doesn't work, but it will always double when Linde is below 50% for brash assault.

Only brave weapons can quad. So no

Only weapons that have the possibility to quad...
+Brave Axe
+Brave Sword
+Brave Bow
+Brave Lance
+Amiti
+Dire Thunder
+Alm's Falchion Refine

I don't think I missed any.

No because there are no Brave Daggers

If you pair desperation with a brave weapon (weapon with a brave effect, see list above), and you meet the trigger requirements and you are getting a follow up, it will work.

So the answer to the question you put in the title is technically yes, but the Janky summer Linde build you described won't quad.

One of my favorite gimmick builds is brave weapon + gale force with anything to give you a guaranteed follow up. Desperation optional. Cordelia already does this with her base kit.

EDIT: looking at your A and B skills, it's also worth mentioning that desperation doesn't trigger in the enemy phase. Wasn't sure if you were expecting it to or not. Dunno if starfish is programmed differently...