The "definition of insanity" solution
There appears to be a glitch in Hearthfire where giving your kids clothes can make them naked. The solution to this appears to be to give them more clothes.
It's possible that an unrelated glitch made them naked in the first place, and then giving them clothes made them naked-er. Giving them another set might be enough to fix the problem.
The "abandonment" solution
Many bugs in Skyrim can be rectified by leaving the area long enough that the area resets. For this to work, you've got to leave the house (and the general area - get far away for best results) and then take it easy for about 10 consecutive days. You won't win any parent-of-the-year awards, but you might get your kids clothes back on them.
The "put that down" solution
If you're playing on PC, you can try emptying out their inventory and then giving them clothes again to see if they'll wear them. As always, back up your save before playing with the console!
- Open the console by pressing the ~ key
- Click on the child.
- Enter the command
removeallitems player - this will put everything in the child's inventory into yours.
- Disable the child with
disable
- Enable the child with
enable
Now try giving them their clothes back, and seeing if they put them on.
Ok.
In Breezehome, there are 2 beds. Your master bed, and a followers bed.
In most other houses, the same thing applies so make sure you have 2 non-owned beds not including the master bed for your children.
Second, if you own the DLC houses, you may want to build a bedroom addon as well as the main halls 3 beds. Because you can obtain a Bard, a Housecarl and a Steward, thats 3 beds taken already. It's basically all based off of free beds.
And 2 children is the limit as-well.
Also, you said you have 2 beds taken up and only one child. Its because your housecarl owns one and Dorthe most likely owns the other.
Best Answer
If you don't have Hearthfire, then you'd have to install mods like Adopt a child to allow adoption of children. This mod, in particular, will allow you to adopt a child from the Honorhall Orphanage in Riften.
I've checked out Skyrim Nexus and it seems that the current available mods there that allow adopting of kids not from Honorhall require that the Hearthfire DLC be installed.