I'd say that 3rd is a call, rather than a raise. You've got a good starter for the low, and a decent, but not great hand for the high. It's speculative, and as such, you want people in the pot, rather than folding to another raise, simply to get more money in should you make a big hand.
4th street is certainly a raise. Your 7 draw is rough, but still looking better than any other, plus the straight draw, even though it is a gutshot, is a welcome addition. I normally would say to put 1 raise in here, but no more, in order to build value without scaring anyone off, but you do actually want to scare off the K-3 so that you can claim the low by yourself. (at least that's how I would play it) Now that you've got a good advantage for the low, make him pay dearly to try to outdraw you that way, otherwise you risk him catching good the rest of the way out and you catching bad (exactly what happened).
By the time 6th rolls around I think all signs point to folding as your best option. You are drawing better than he's made, true, but there's nothing to say that he can't catch better, even if you do too. Even if you make your 7 low on 7th, he is drawing smoother, and thus can hit a 6, 7 and either an A,2 or 4 (depending on what he's got in the hole, and considering his raise on 3rd, I'd say he's probably close to perfect in the hold).
So you DO have a few outs to win the low, but he looks to be very strong in the hole (I'm thinking 2-4 in the hole) so he has a ton of outs even if you do outdraw him, plus, if my guess is correct, he's got an open-ended straight draw to possibly scoop with. So you're a big dog both ways. Not a good spot to stick around and chase a low that still might not be good enough.
I would definitely say that your only option on 6th in this situation is to let it go.