Even if there was only a few inches below the joists I bet that rigid foam sheets could be shoved under for a pretty good job - even if it was just resting on the dirt. Might not be perfect but a ton better than what you have. with Rigid foam you don't have to worry about a vapor barrier b/c it doesn't need it. Be careful if you do decide to have someone dig it out a little to make sure they don'e undermine the foundation footings - these hold up your house. I would never consider ripping up everything to insulate. There has to be another way, call several insulation companies and have them come over and give you a FREE written estimate. You probably want the guy who doesnt' just sit there and scratch his head, but you DO want the guy who seems to really know how to solve it and has done it before.
What is the insulation in the walls and ceiling like? While the floor is important, it is not nearly so as the other parts of the room. Typically, floors account for about 15% of the heat loss - heat rises for the most part. You might be able to make the room more comfortable by addressing the other areas of 85% of your heat loss. How are your windows and weatherstripping? Air infiltration might be accounting for more loss than your floor.
Just because this is the only part of your house that is over a crawl AND is the only cold part of your house, does NOT logically PROVE that this is the cause and that insulating the crawl will fix it. These two factors are "associated" but not necessarily proved to be the causation. It COULD prove out that you go ahead and insulate the crawl and the room is still cold. You should insulate the crawl but you need to look at ALL POSSIBLE aspects of the problem.
Traditional open hearth wood burning fireplaces are usually not very good at heating. Modern builders usually put the chimney on the OUTSIDE of the house instead of in the center of the house and this causes problems too. If you are using central heat (a furnace - which you ARE) then when you start a regular fireplace it actually sucks the air that your furnace has warmed right out the chimney. Starting a fire in there probably loses you more heat than it provides. Make sure your chimney flue is sealing when not in use or that could be part of your heat loss problem.
You need to have heating contractors advise you on analyzing and possibly improving the heat delivery to this area. Remember, if you have an attic space above this room then maybe ducts/pipes could be run there also. I have lived in houses with NO floor insulation that were bearable (just) because it had a lot of heat delivered to the area to make up for it. As to your last question, I don't think there is any good reason to heat the crawl space. This is not living space and dumping heat into it would be largely wasteful - if you are going to dump heat somewhere then dump it into the room itself - the actual living space.
If you are considering improving the fireplace then you have a lot of options. First of all, I would FORGET about the vent free options, they are supposed to be decorative and not for real heating. 1. Leave as-is but have some glass doors installed that are fairly air tight. Your fire will burn fine and the doors will prevent 98% of the air/heat loss that was happening before. You might boost your heat also by using a heat reflector behind the logs. 2. Accomplish the same thing and more by using a manufactured insert or stove, either wood or gas, either with or without a fan. Have installing contractor and fire inspector decide how to treat chimney flue, i.e. stainless liner or not, most likely YES.
For my strong feelings AGAINST vent free for heating purposes see further down on the board: http://184.108.40.206/wwwboard/messages/51478.html and my follow up to that post. You have a chimney right there so why wouldn't you use it?