I don't think the crowd noise would be too problematic for residents. If those buildings were all residential, the closest would be over 900 feet away from the stadium. For comparison that's similar to the distance between Fenway Park and Timeout Market, the Kenmore MBTA station, and Queensbury St. Not exactly right on top of it. Decent construction and soundproofing materials should further mitigate the impact. Besides, you're looking at a max of about 14 games per year (inc. preseason and playoffs during an AFC Championship run) vs. 82+ playoffs at Fenway. Revs crowds are about 1/3 (or less) the size of Patriots crowds, and a handful of concerts each year shouldn't be too problematic.
You're right that ample parking would be essential for any residential development in Foxboro. You're most likely not ditching your car(s) if you live in Foxboro, even in a development like Patriot Place. I'd also wager that it would have to go hand-in-hand with some longer term guarantee of continued Foxboro commuter rail service. If I were looking at that development, my big concern would be traffic on game/event days. I think that would be more prohibitive than the potential noise. But the idea of throwing my place (and parking spot) up on Airbnb for football games or Taylor Swift concerts would be extremely appealing. I would wager a 1br condo there could easily get $1k+ on a game night or big concert night.
