First and foremost, should Andrew Garfield be the Spider-Man to join the Marvel Cinematic Universe? My answer is yes. By the time MARVEL get's the rights back, a lot of the actors will be older and it will be time to reboot the MCU.
This MARVEL Cinematic Universe is probably the best thing that could happen to CBMs, and seeing Sony and Disney put corporate greed aside to give fans what they want, would go down in cinema history.
The possibility of it happening is mostly shown by the fact that Kevin Feige hasn't crushed nor confirmed the rumor yet, he's obviously helping people board the hype train.
The question is how should they do it?
Well here is my idea. I've shared this idea before, but for this article I've made major alterations
Primarily, TASM should take place after IM3, this would explain all the cranes, and explain why Iron Man wasn't helping to fight the Lizard.
First, combine the universes with a lot of subtle references (To catch the comic book fans) and some very direct ones (To catch the general audience) in Amazing Spider-man 2. Be it a headline about the events of a MCU movie in the news paper, or just someone mentioning the name "Stark".
This shows that they are in the same universe, but just haven't interacted yet, plus this is how MARVEL combined their cinematic universe before Avengers came out.
Now comes the hard part, introducing him into the MCU without the angry fanbase saying he became an Avenger too early and that it doesn't make sense.
Here we go, yet again we have to take steps, just like when they built the MCU. Put him in the second Avengers, like Thor have Spiderman not be part of the Avengers initiative, but instead CAREFULLY TAKE AIM and throw him in some how. By carefully take aim, I mean have it make sense.
I believe the easiest way to do this is have a battle take place in New York mid-movie. In TASM2, Spider-Man could swing his way in and kick a henchmen who is about to shoot the ever-living crap out of Thor from behind.
After we get an epic action scene in New York, spend some time developing his character, true he is a lot like Tony Stark, but he sees the world differently, have him argue with Tony about how it's his responsibility to protect his loved ones without need for publicity.
Finally, at the end of Avengers 2, he is asked to join the Avengers, but refuses, saying a speech SIMILAR to the following.
"You guys are this planet's soldiers, you guys have the responsibility of protecting the Earth, but me? My responsibility is in New York, to protect the ones I love and care about. I wear a mask to make sure they aren't harmed. Unlike you guys, I still have a family and friends to care for, and they are my top priority. Besides, after seeing you guys in action, I, for one, know you don't need me."