Illustrator :: Exporting Image To PNG?

So, I'm developing a game currently and have been drawing some character portraits (busts if you will) in Illustrator CC (v17.0.0 32-bit).  After finishing each portrait, I need to export them into a .png file that I can then use with MSPaint.
 
My troubles are that I always run into one or more of the following problems each time I export an image:
 
If I save the image at 72ppi, the exported images are the correct size but, they have jagged edges, which just aren't acceptable for my project.If I save the image at say 300ppi, the exported images are blown up and are too big but, they have smooth edges, which, while I want smooth edges, I cannot work with the increase size of the images.Sometimes exporting the images causes a new line around the smooth edges (I think I've heard this effect called a Halo or something along those lines), which is also unacceptable because I need a transparent background.Also, if I resize the blown up image manually, it causes develops the third problem. 

What I'm after basically is to be able to export the image on my artboard into a .png file, have export at the same width and height it does when I export it at 72ppi, while keeping the edges smooth like they are when exporting it at 300ppi.
 
As a side note, I currently don't have a suscription/access to photoshop and don't have a way this month to gain access and though there are freeware programs that do a lot of what Photoshop is capable of, I can't seem to figure them out.  I've mentioned this just in-case potential solutions require photoshop