You asked about the choice of mat. I'd use 1.5 oz/sq. ft. mat in several layers. The total that you mentioned of 6 oz of glass per sq. ft. is reasonable, but you want a reasonably thin and flexible mat in order to make it around curves and corners without getting excessive springback, in my opinion. Thus, better to arrive at your 6 oz total in 4 layers than trying to get there in one. Also, a mat in the range of 1.5 oz is reasonably easy to wet out, and anything much heavier will get increasingly difficult. You can also include a layer of glass cloth or woven roving if you want to add a lot of extra strength, but its not typically necessary. If you do decide to use an alternate fabric, be sure to put a layer of mat on both sides of it.
Be sure to use a serrated roller, they're not that expensive and very helpful.
The preferred type of resin to use for this would be an isophthalic-based polyester. If you can't conveniently find one, a general purpose resin will also work, you'll just be giving up a bit of heat distortion temperature and a bit of strength properties. Probably a more critical thing is to get one with a reasonable gel time- using one that is very fast, as might be used for chopper gun applications, will have you spending all your time mixing up very small portions and the process will get very tiring in a hurry.
I agree with cboy's comment that warpage will likely be an issue for you. It's caused by the fact that these materials will tend to have some shrinkage as they cure, even though that shrinkage might be relatively small compared to other types of resins. You may be able to reduce the warpage some by first applying a single layer of laminate to both sides of your cardboard before building up successive layers of laminate, if possible. Anything that you can do to brace your cardboard so as to help control warpage would probably be well advised.
One thing that I differ in opinion with some previous comments is, I would not allow a first coat of resin to gel without putting glass mat in it! I'd simply brush my first layer of resin onto the cardboard and then proceed with laying the first layer of laminate into it and proceed with wetting it out with your serrated roller. A hand layed laminate is almost always on the resin-rich side unless your using an exotic technique. I think your risk of getting cracking on that layer of resin that wets out the cardbord in the absence of mat outweighs any other considerations. This is my opinion based on theory only, as I do not personally generally laminate over cardboard- perhaps someone else on this board with experience in doing so might want to comment as well.
You mentioned considering using tiger hair or kitty hair on the backside of the cardboard. I'd far prefer to simply use glass mat and resin, with the exception being if you have certain areas that have sharp interior corners or tight curves that would tend to give you springback problems (the layer of glass mat springing away from the corner rather than conforming to the curve) then a glass-filled putty like tiger hair would be an excellent choice to use in those areas. Just make sure you get your putty or tiger hair layer as smooth as possible, so that your laminate won't have trouble covering it evenly without leaving voids.
Having said all this, the real work involved in this project is going to be getting the laminate smoothed down to give a smooth surface on the final appearance surface. That's the reason that most laminates are layed into female molds, with the surface that contacts the mold being the final appearance surface. My suggestion would be for you to make up a relatively small "test" layup on a trial part to make sure that you're going to be satisfied with the final result, and comfortable with the amount of work required to achieve your final appearance, before you begin laminating over the entire body. The alternative approach, of course, is to use putty over your body to create a male buck, then laminate a female mold over that male buck, then finally laminate your actual part into the female mold. It's a good deal of work either way.
Best of luck on your project, and please post some updates of how you get along!