I'm trying to install a ceiling fan, however I'm not sure if the electrical box is rated for a ceiling fan. There are no screws available in the box. Am I supposed to use the screws through the box ears and then into holes at the end? The holes at the end only seem to have 2-3 threads, would they be enough to hold the weight of fan? The fan I'm trying to install needs at least 35 lbs weight (actual weight of fan is around 15 lbs). Do I use 10-32 screws or 10-24, and what length?
The label says so, so yeah, it's fan rated
You can see a partially painted over label there that says it can handle a 70-pound fan or fixture when attached to joists that are 24" on center, so yeah, it's a fan rated box alright. However, it appears to use a metal hanger bar running across a joist bay (instead of mounting to a joist directly in the fashion a pancake or saddle box does).
As to what screws to use....
I would use 10-32 screws, about 3/4" longer than the depth of the box (so 2.5" or 3" long) and screw them into the holes in the back of the box that lie behind the cover-plate mounting ears -- they need to engage into the hanger bar behind the box in order to mount the fan correctly, it appears, in your case.
Study this installation instruction sheet: Arlington FBRS415 pdf . There were two screws that were originally "parked" in the box to secure the fan bracket to the "ears" and those screws have been removed and lost. Contact the mfgr and get some new screws, if you want the exact type originally supplied. You could buy a complete new box of this type and remove the screws from it, but there are fastener outlets from which you can purchase specially rated screws.
But for the light weight fan that you are installing I believe any sufficiently long screws of the right diameter and thread pitch would suffice. However, if later someone wanted to put in a very heavy fan, then ordinary screws might fail. The rating of the box surely depends on having the mfgr supplied screws or equivalent.