The double speed increase will only come into play when you're below your resting point, so it won't be a factor once you're allied. It also won't matter if enough time passes after you protect or pick up the patronage policy and before you start gaining influence actively. Even then, that part is just a smallish lump sum unless you're constantly doing things to make the CS mad.
Though you'll get a bonus for all your allies, the AI won't compete with you for all of them. Even without Greece's UA, once you're allied with a city state you'll get enough freebie influence from quests to stay allied. Since 250/60 allied is the same as 80/60 allied, the UA only matters for the CSs that the AI is competing for.
Bonuses to your gold/influence ratio from reformation or patronage actually make the Greek UA worse. By decreasing the gold cost of a point of influence, they decrease the value of a free point of influence. Interestingly, a civ with a gold generation bonus is going to have the upper hand on getting the bonus from those first few city states sooner compared to Greece. So if you're concerned about any multiplying effects of getting some nice happiness or food or whatever bonuses early that's another mark against Greece.
It's harder to say anything about the effect of larger map sizes where there could be 20 CS. Obviously it increases the max potential value if you're allying every city state on the map. Some of the money-friendly civs have bonuses that scale up with map size and some don't.
That is not exactly right. You see an 80/60 influence can be overturned with a reasonable gold investment and a spy. A 200+/60 is nearly impossible to shift and demands a huge amount of spy effort and treasury dipping. Also you forget that your influence falls down every turn even with religious and policy fail-safes. If Alexander gets the same fail safes his own will stop degrading completely. So in effect the only loser out of this situation is the one who is trying to take the CS from Alex.
As to quests, you cant be everywhere at once. Not all will be readily easy for you to complete and everything else will require 30+- turns to complete due to the inherent nature of them. Not factoring in the impossibility to complete certain ones if you dont have a lead (culture for example).
So all in all what I am saying is this: Since Alex prioritizes city states, most of the time he will beat you to them. A smart human player will have them in his grasp by mid game. Considering all of us go rationalism as soon as possible he has a big head start on influence by virtue of not degrading. When Patronage (which IMHO is not even remotely necessary for him) and Ideologies hit the table he simply seals the deal. You need extreme expenses to compensate with a top notch Greek empire in the CS war. In fact most of the time a conventional war is the only answer.