Hydra's compromise of SHIELD from the beginning is dealt with at length in Agents of SHIELD, but it's pretty open in The Winter Soldier and Civil War.

As for Lawson, I thought she was Kree. It's possible I missed that because I didn't find Captain Marvel all that memorable, but I did think she was evil. In any case, agree that she was experimenting and that it ended up on the ship before Goose "gave" it to Fury. I don't think that's inconsistent with The Avengers because of the different time frames.