I think we all need to relax here
You know guys, not every kid looks like their father. I know several people, including myself who don't look anything like either of their parents. It's really not that important a factor, especially in movies.
As far as Shia's character goes, we have basically seen him in teenage roles set in modern day. Who knows what he has in store when he actually has a play a real part that requires him to act different. Plus, has Steven Spielberg as a director. And how cool was that dagger stuff he was doing? It looks like he worked hard to develop an interesting character and not just some teen punk like usual.
Honestly, I'd be ok with it if he got his own franchise (as long as he was a different character). If I didn't want to watch them, I'd just ignore them (much like I do Star Wars Eps. I, II, III) and still enjoy the originals.
And about the hat thing, people aren't thinking in terms of Indy's character. Sure, the hat means something to the filmmakers and to the fans, but does it mean anything to Indy other than just being a hat that he likes? My point is, I don't foresee a scene where he "passes the fedora" on to his kid. It's not in Indy's character to be so grandiose about himself. And putting on a fedora doesn't automatically make you an adventurer anyway. I just don't think it would occur to Indy that the hat is what's important (other than something to grab before the descending stone door closes forever). You all know what I mean?