Fallout – Can’t rearrange vault because of Living Quarters/dweller limit

Mating: Bring a male and a female Dweller together into the living quarters. In larger living quarters you can bring in more Dwellers and let them work it out. The higher the Charisma the faster they will mate.

Direct offsprings will not mate, the game prevents incestuous behavior. If you do not keep log of your bloodlines (which you most probably don't) there are two ways to detect blood relationship:

1. By default new borns will get the last name of the father. So if two dwellers have the same last name, they might be related.
2. In case related dwellers are in the living quarters, they will bring up the topic that it is nice to meet with the family.

Pregnancy: While women are pregnant they can work. But they will not take care of any incidents but run into the living quarters instead. Make sure to assign at least one male/non-pregnant Dweller to all stations and don't equip pregnant women with weapons since they don't use them.

Deliver Baby: After mating it takes 3 hours for a baby to be born. You do not have to be in the game for this whole time. Just come back after 3 hours and you'll have a baby.

Make sure to have enough room in your living quarters. If your vault is full, women will simply stay pregnant until there is free room. There won't be an indicator and neither the pregnant dwellers will complain in any way. (Also see maximum population below)

Growing up: It takes another 3 hours until children are fully grown up. In their stats they have a grow-up-bar. During this time they can not be assigned to jobs. Again you do not have to be in the game during this time.

Overpopulation: Take care to not grow too fast. Once a baby is born you will need to feed it. Since children won't work you quickly run out of resources and have to juggle your workforce between your water treatment plants and diners if you get too many babies at once.

Maximum Population: The maximum number of dwellers in the vault is 200 - if you reach 200 dwellers, you have to be very careful placing men and women together in living quarters, or you'll have a whole lot of perpetually-pregnant females.

I've noticed that deathclaws tend to attack right after a new dweller joins the waiting line outside my vault OR right after I let someone into the vault from the waiting line. They don't seem to attack randomly while nothing else is going on.

To answer your specific question:

How can I stop all these deathclaws from eating my dwellers, while still attracting new ones?

You can't stop them

You can stop them from coming by not clicking on the radio room (so no one gets added to the line outside the vault) and by never letting dwellers in or out of the vault (so the door never opens). Obviously this is not really feasible as it prevents you from adding fresh blood to your vault and from getting the weapons you need to defend it.

In case you thought lunchboxes would be your savior: sorry. Dwellers from lunchboxes don't bring deathclaw attacks when they join the waiting line, but they can still trigger an attack when they enter the vault. I haven't done testing to determine if there is a lower percentage chance for lunchbox dwellers, but I have triggered a deathclaw attack when I brought Bittercup into the vault.

But you can resist

You can minimize the damage that the deathclaws cause by arming all of the dwellers in the topmost levels with the best weapons and outfits you have. Posting guards in the vault entrance is also helpful. Your goal is to have each room wear the deathclaws down as they go deeper into the vault. The deathclaws move from room to room fairly quickly, so manually sending a dweller with the best guns to chase them down is difficult. You can try sending the defensive guys a few rooms ahead of the deathclaws to wait for them, but equipping the the dwellers assigned to the room with guns is faster and should have better results.

Plural of anecdote is data, right?

Some facts from personal experience:

In my fancy vault with 150+ dwellers and Mr Handys, the deathclaws will be killed on the second or third level down. In that vault, everyone on the top floors is carrying weapons that do 10+ damage per hit, and the deathclaws still get past vault entrance, top floor room (with 6 dwellers), and one or both of the second floor rooms before dying. No one dies when this vault is attacked, but they whine about the damage they took.

In my not so fancy vault with 100 dwellers, everyone on the top floors is carrying rifles or better (so minimum 5 damage per hit on most of the second floor and third floor, min 6 damage on the topmost floor). Deathclaws here will get through the top two floors with ease, start to die on the 3rd floor, and be finished off on the fourth floor. After the last attack I had to revive three dwellers and pass out all 25 stimpacks. The deathclaws also stole/drainined/rekt about half of my water/power/food resources. This was a pain in the ass but not as difficult to get out of as the early vault death spiral.

tl;dr:

Don't let anyone in or out of the vault; don't let anyone line up in front of the vault. Complain to Bethesda so that deathclaws will get nerfed in the next patch.