Sure. What you're talking about is similar to the "Electrician's Outlet"... an outlet right next to the panel which is fed by a dedicated breaker. So the electrician can plug in portable lights etc. when the rest of the house is turned off for maintenance. That's pretty much standard practice for interior panels.
The electrician's outlet just uses a conduit "nipple" (prefab short pipe) that gives you an inch or so of length, so you get far enough away from the panel so the covers don't jam.
One common trick I see is using an offset nipple, which is about 2" long and has a small zig-zag. The offset isn't used for anything, it's just a handy, pre-made 2" nipple lol.
But if you want to run conduit a little farther and make it 3' away, that is fine. I recommend planning on water getting into the pipe, and exiting the bottom of the panel in a "drip loop" sort of deal.
Any outdoor recep is going to need an outdoor in-use cover.
You cannot use Romex (actually NM-B) in conduit because Romex (NM-B) is not rated for outdoor use. Also, there's a size issue: you must treat any cable as a single round wire of the large diameter (because it twists), and that makes it too big for many conduits. UF cable is allowed outdoors, but it's even thinner and wider than NM-B, and that makes it require HUGE conduit!
I'm not really sure what people's hangups are with THHN, it's really, really wonderful stuff, especially stranded. You'll never want to alligator-wrestle Romex again! The only snag with stranded is it's hard to hook over screws, but GFCI receps have screw-to-clamp terminals, and that makes it easy.
If you run #12 wire you are welcome to use either a 15A or 20A breaker as long as it's a dual 15A recep. (a 20A recep is also acceptable on a 20A breaker).
All the THHN I own is 12 AWG (except for the larger stuff obviously). I don't see any reason to own it any #14. #12 works for #14 and now I only have to stock one thing.