What type and length of screws should be used to attach a plate caster to  3X3  post legs?

 Personally I would go with lag bolts. You will get a better bite in the wood.

1/4" should be sufficient if you are doing 4 per caster ...1 3/4 or 2" 

what's the size / estimated weight of your box?  I'm not sure about the size, but I STRONGLY suggest drilling holes that are smaller and just a little shorter than the screws you end up with.    I'm a VERY amateur DIY man, but I've NEVER used lock washers in wood.  Probably wouldn't hurt anything, though.  

2" lag bolts the diameter of the caster holes, with a lock washer, and some wood glue for the drill hole, if going into the cut end of a post. .Posts need to have holes drilled more than half the diameter of lag screws to avoid splitting the post, when they are installed. This is the forever method.

Or you can use 2" grey deck screws with a washer (because the heads are small.) The yellow ones are cheaper, but they rust more quickly. Deck screws are not gonna be nearly as sturdy with a bed full of dirt.

thickest ones that will fit into the holes by 2 inches [50 cm] long

That's a lot of weight so tge casters are heavy duty. I would estimate #10 Wood screws by 2 inches. 

Bring the caster to a hardware store and take pictures of the legs with your phone.  The hardware store staff will give you the right screws.