No, the tire may be to narrow or too wide for the rim.
550 is 5.5 inch wide tread. 600 is 6 inch wide.
No
31 x 10.50 x r15 mounted on an eight inch wide wheel
Yes but it balloons the tire. That means the tire tread will not be evenly driven on. Causing awkward tire wear.
Rim size is important since you can't put a sixteen inch tire on an eighteen inch rim. But tire width is important too. Don't try to put wide tires on narrow rims. If you have questions, your local tire professional would be a good resource. Perhaps he/she would be impressed that you're a "chick".
square and about 1/4 inch wide
No it will not work. Very dangerous to even try.
For a 12.50 inch wide tire, the Tire & Rim Association approves a rim width range of 8 1/2 inches to 11 inches.
Its too narrow of a wheel. I have a 12.5-20 Michelin Xl and it is on a 10" wheel . It is the limit on that. I would have to say that a 12" wheel is the Minimum for it to fit...
31.6" tall and 10.4" wide.
Can be quite a difference there, from 400-500 grams for an XC tire to almost a kilo for a wide DH tire. Narrow road bike tires can be even lighter.