I was hoping that by sealing/painting the 2x4s that are to come into contact with the ground I can eliminate the need to buy pt lumber.
Sorry, but nope. Painting or sealing the parts on or in the ground may buy you an extra six months, is all. Really not worth the bother.
If I use glue and half lap joints can I get away with using 2x4s. I know I will have to use more than if I used 4x4s on the corners, but I think I can make it work, right?????
I am not clear on what you're proposing... are you saying that your run will not have anything sunk into the ground at all? If so, you *can* build it without 4x4s, yes. If you are not using very strong wire mesh (e.g. heavy-gauge welded 1x1) I would add some diagonal braces here and there to prevent racking. You will also want to give some attention to anchoring it so that it can't blow over, and something (possibly a wire skirt) so nothing can dig under.
BUT if you want to be taking this with you in a few years, I would really very strongly recommend using pressure-treated stock for the horizontal pieces along the ground (which you will need if you have no posts sunk into the ground). If you do not, there may not be much down there *left* to move when the time comes, and believe me it is really really obnoxious to remove rotted pieces of wood from a structure like that and replace them. Another thought, for future portability, is to build the sides and top of the run as separate panels -- sort of like you'd make a stud wall of a shed -- and bolt them carefully together in a way that is sturdy but disassembleable.
OTOH if I've misunderstood, and you are going to sink posts into the ground, you really ought to have pressure treated 4x4s for the corners. For strength and stability of the whole run. If you are dead set on substituting 2x4s, the thing to do would be to nail(or screw)-and-glue two 2x4s together along their full length to 'make' a 4x4. If you have plenty of free 2x4s I guess you could do this... HOWEVER, it will rot through faster than a 4x4 would. And if it's not pressure-treated stock, it will rot thru pretty fast where it's sunk in the ground.
Hope this helps,
Pat