With 7's pre-flop, I definitely would not have limped if I were going to play that hand at all.
7's don't represent a "bad" pre-flop hand, but they certainly don't represent a great one, either, and your lack of a raise allowed six callers to potentially hit an over-pair, straight/draw, flush/draw, etc... on the flop.
If you're going to play a marginal pair pre-flop, I think it's important to protect that marginal pair by weeding some of those other players out with a raise. You don't want to let those guys see the flop for only the cost of the BB and hit an over-pair on you. Not only will raising there represent a premium hand, allowing you to bluff should any paint hit the board on the flop (especially if you hadn't hit your set), it would've also allowed you to bully any would-be pot-stealers if they'd limped and hit the high-pair, but with a poor kicker. For example:
If someone had hit that king on the flop but they knew they had a weak kicker, your pre-flop raise could've represented that you also had a king, but with a better kicker. They may have even thought you had 2 kings in the hole. They may have figured you for AK, or KQ. Either way, if they're staring at K5 and realizing their kicker is actually coming from the board, and not their hole cards after you've represented something premium pre-flop (masking your true hand, the 7's), at the very least you're putting a seed of doubt in their mind, making them worry that their weak kicker wouldn't sustain them if it came down to a heads up situation. You don't want to lose a hand because you were afraid to protect your hole cards.
Since I don't know what the blinds were after the flop hit (and therefore, what the overall value of the pot was at that time), consider this an observation more than a critique--
Once you saw the flop, realized you'd hit your set and only bet $5 (after limping pre-flop) you priced the other players in, increased the risk you'd get sucked-out on, and undervalued your hand while minimizing potential return. I understand that you were trying to set the hook and keep these guys in the hand, but it's important to get some added value for your set. Premium hands don't come around often enough for any of us (I'm sure we'd all agree) and when we see them, the point is to get paid-off. You're lucky you forced three players out of the remaining six to fold with that play. If all six had called your $5, that's 5 players that could've been chasing open-ended straight (with the 4 and 7 on the board) which would dominate your set.
However, if the turn were another king, the person I mentioned previously who hit a pair of kings with a poor kicker on the flop would then have a set of kings, but you'd then have a full house, potentially increasing your return.
It's a risk/reward thing, but if anyone had had "5 6" in the hole, the chances of hitting an open-ended straight after that is 5-to-1, which is much higher than the chance you'd hit a full-house. You need to leverage the odds as much as possible, and if you're going to string players along, encouraging them to hit something on the turn that is still weaker than your hand, you need to be damn-sure that you already have the nuts, have an extremely high probability of hitting the nuts on the turn/river, or have something that may not be the nuts, but is the second best thing.
In conclusion: I personally believe (and I'm almost positive Doyle mentioned this in "Super System 2," though I'm not positive on that) if you have a marginal pair pre-flop and are going to play it, you need to raise to represent something big, earning you leverage for continuation-betting on the flop, turn, and river. Further, if your'e lucky enough to hit a set on that marginal-pair when the flop hits, you need to protect that set by betting something you can be confident will weed out the stragglers who are chasing something on the back-end that will dominate you.
You're lucky to have won the hand (you DID win the hand, right? or am I reading that wrong?) because anyone could've hit a straight given how many folks you let come with you, and because you didn't protect your hand by increasing its perceived value, you left the outcome up to chance, and you should limit that as much as possible with smart play and skill.