What is the difference between 'futari' and 'futari-domo'?
Also, is 二人共 the correct way to write 'futari-domo'?
Yo Hamsteriffic, you probably mean futari-Tomo.
The difference is as follows:
二人 ⇒ two people.
二人共 ⇒ both people.
“二人” means two persons, e.g. 二人で合唱する ‐ Two (people) sing a duet.
“二人とも” means both of the two, or two of them together, e.g. 彼は兄弟二人とも東大出身である ‐ Both he and his brother graduated from Tokyo University.
“二人共” can also be written “二人とも”, and should be pronounced “Hutari tomo,” not “Hutari domo.”
When we say 共々, e.g. “夫婦共々”元気に過ごしています ‐ "We, as a couple, are faring well," it should be pronounced “Huu-hu tomodomo.”