Swarm

Right. Sounds like a plan. Ideally I'd find a frame that's got brood in all stages, capped to eggs.

Did the swarm arrive in the trap themselves? Rare that they would be without a queen, but not rare that you wouldn't be able to spot her, especially if she's a virgin. If they arrived on their own, leave them alone for another 3 weeks. If it's something you harvested, give more info. And to answer your question, yes, you can add frames with eggs...but may not be necessary depending....

Yes do that and the next day you will know if they have a queen. (it still could be a virgin that needs to mate)

No emergency q cells = they have a queen.

Last night I took 1 frame of capped and uncapped eggs from my strong hive there were some bees on the frame when I added them in the nuc box with the swarm, I screened off the entrance so they can not get out and put a feeder in 1/3 honey 2/3 water they drank about half . Tonight I was going to take the screen off and reduce the entrance after dark, and continue feeding for just 1 day and then taking the feeder out tomorrow night.
I forgot there is no Queen in the swarm it's been a couple days now that I have been checking and no sign of her. Tomorrow will be 2 weeks for the swarm in the trap

There had to be a queen or there wouldn't have been a swarm. Virgin queens are smallish, not much bigger than a worker. She may be in there but she's hiding from you. If there isn't a virgin or mated queen now but there are eggs and small larva on the frame you added, you should see queen cells in a day or two. If there is a virgin or mated queen you won't see queen cells. Flower planter makes a good point.