Oh dear, this does sound awfully stressful for you. I sympathise as have frequently been piggy in the middle with DD (now 18, away at uni) and DH. He is also an old fashioned type, though not as much as your DH sounds (sorry!). We/they fell out over the years about short school skirts, make up at school, not working at school, sex at 16 (in a 2 year relationship!), more recently an unsuitable bf (he was, mind!) and tattoos. Such fun! So yes, I've been where you are. They get on OK now though. I have to admit it probably helps that she's living away during term time, but things did improve last year when she was still home.
I had lots of talks with both of them, individually. Talks, not rows. Catch your DH at a good time, make him see reason. He surely knows, whatever his beliefs, that most teens are having sex at 19 and a heck of a lot younger. Her bf isn't flaunting it in his face, he's just being normal. Maybe you could get him to see and accept this? My DH also used to get angry about her attitude and not helping much about the house, and I could see his point but she wasn't going to have a good attitude when she hated him, quite frankly! I know she's 19 but he still holds the power. She's still in 'child' role in some ways, so I think it's up to him to 'give' first. To my DD's credit she often acted the adult first, and he responded well, but I think it should be him ideally who 'makes the first move' in making peace. He must love her, and he must love you, work on that when you're discussing things with him. Surely he would like a better relationship with her?
If the chips are down though, bugger him and back her. That's what I did. Not about stuff like housework or nicking the car, obviously, but the bigger stuff. He was fine with me, and distant with her. But over time he backed down. What's the worst that can happen? He'll sulk for days? So what? I keep reminding him she's a young adult and her choices aren't purposely to get at him, and she's also very like him and that's why they clash. And also THINGS COULD BE A HELL OF A LOT WORSE!! He should look at the positives.
As for DD, I told her we'd back off and let her have her freedom. Everything improved from that point. We relaxed. She relaxed. The helping at home started to happen. We got her back, basically, and she eventually dumped the bad news, abusive bf.
One last thing I once did was on impulse take them both out for tea and say things were awful at home and we couldn't go on like this and what were we going to do? What bothers him (let him speak). What bothers her (let her speak). What can they do (and me) to improve things?
Sorry for the long post! Good luck 