Random thoughts:
First, welcome to the forum . . . you're in a good place if you have questions about heating with wood.
As mentioned by others . . . even free wood isn't totally free . . . not counting the cost of gas to transport the wood from Point A to home or the fuel for a saw . . . or the cost of buying those "must have" toys such as chainsaws, woodsplitters, mauls, axes, etc. . . . there is the time that it takes to process wood . . . and for some folks time is money . . . or in some cases more valuable than money.
You have a bunch of strikes against you with a basement install unfortunately . . . Strike 1) A lack of insulation in the home. Strike 2) A basement install in an uninsulated and unused basement away from the main living area and Strike 3) As mentioned having a bunch of holes in the floor could be a violation of fire code . . . and "everyone else does it, why can't I" didn't work with my mother and it may not work with your insurance agent or local fire inspector . . . or it might . . . best to ask first . . . and doing so may be OK . . . but it isn't the safest thing to do.
That said . . . there are folks here and I know folks who have basement installs . . . sometimes due to lack of space, sometimes they figure it's a good way to get it out of the way (which blows me away since I really like watching the fire, smelling the potpourri and listening to it crackle away) or that's where the easiest install location is . . . it can be done . . . but just bear in mind you will be losing a lot of your heat which translates into wasted wood which translates into having to cut more wood which translates into more work.
I know you say space is at a premium . . . but I would really look around at the various options . . . many stoves have pretty tight clearances . . . and when done right a stove in your living space not only heats you and your home better by being where you spend the bulk of your time . . . but you will find yourself drawn to it. In my own case I have a decent sized living room, but my wife was still reluctant to give up too much room . . . our compromise was a corner install . . . and we took down a non-load bearing wall and moved it back a foot or so.
Safety of the children . . . there is a recent thread on this very topic. The general consensus is that gates can be bought or made cheap enough . . . and most children learn very quickly to stay away from the stove . . . or you can invest in some of those shock collars . . . and yes I am kidding.
If you opt to not go with the wood furnace and want to go with the stove in the basement . . . I would go larger since you will be losing a lot of heat.
Best advice . . . and it has already been said . . . but it bears repeating . . . in fact it was some of the best advice I took away from here when I was researching woodstoves . . . before you spend money on a woodstove it's almost always better to spend the money on insulation . . . preserve the heat you generate whether it be from wood, oil, gas or electric . . . otherwise that heat you bought (in time or money) will not keep you as warm for as long . . . on the otherhand you will be doing your part to help heat up the neighborhood and the great outdoors.
Seriously, before I bought my stove I spent a good portion of my heating funds on insulation to trap and preserve the heat I generate . . . whether it be from the oil boiler or woodstove.