In what ways would I be able to take links out of my mountain bike chain, without a special tool?

The "figure of 8 for the gears" is your rear derailleur. If you remove it you will no longer have any gears on your bike. Does your bike have horizontal dropouts? Probably not, so your chain would likely end up being too long and prone to falling off. Even if your chain can be cut to the right length, which gear do you plan to use? Most of the time it will be too hard or too easy to pedal. Once the chain is cut you can't go back. To cut and rejoin the chain you have to spend the money for a tool. perhaps it might cost about the same or even less to repair your derailleur and tyour gears would work again.