I'm running a pbp game - so no tactical movement, and only a vague map. The party is entering an area and is likely to trapped.

Mechanically, how do you run a game to account for:
1) The random chance of hitting a trap
2) Detecting the mine once you've stepped on the trigger so you can sit there and sweat
3) Determining damage for non-conventional explosive devices, like exploding gas cans and the like.

I use SR3/SR3R, so damage codes and skills need to comply with that (but I don't mind being a little more lenient in other aspects, like odds of walking across a trap).

Well aside from making a detailed map...

I would randomly determine who triggers the trap. Assign each player a numbe or two and roll a die. Allow that PC a perception test and a Quickness test to see of they notice and/or respond, respectively. If not "click boom".

If the trap is in an area they will all traverse (a hallway for example) then whoever takes the first action has to test.

Who builds a trap which only covers half the entrance?, or where the pressure plate is shorter that a wide-trollstride?

No dice, as a gm i just let them roll perception a few time at the start and ask about in what formation and distances they normnaly "travel" after that (if no one sees it) i DETERMINE who activates what.

The traps are set in an open air area, so not restricted to only passageways.

There you have it. You either have to map it out and track movements in detail, wait for a player to vaguely describe moving into the area of the room that is boobytrapped or assume they mill around in a general sense in which case random determination is the only fair way to proceed.

I guess a 4th option would be to "boobytrap a plot element"- i.e. whoever finds the mcguffin gets the shaft.

A person enters randomly in the area where a trap is. How do I determine if he's unlucky enough to trigger the trap? Do I need to divide the area by 3x3 blocks, and roll randomly to see which he wanders through? Just say "hey, you trigger one"? How do you determine successfully crossing a minefield?

If you don't want to draw it out and record movement then you're probably going to have to do it randomly. Doing it randomly you could have them all roll their edge attribute. Sounds like a luck situation to me. Could be a threshold or if it's important that someone trigger it you could use the lowest score(then lowest pool on a tie) or something.