Your explanation of tonic, supertonic, mediant, I'll accept. As far as I'm concerned, 40 years writing, it's useless information. Nice questions on a test.

As for tritones, I don't think you're gettin' it. The tritone is the Maj 3rd and the b7th of a dominant chord.
The interval is a tritone. Or a #4 or b5 or the devil's interval. A fully diminished chord is a double tritone. e.g., D7 and Ab7, both chords' tritone are exactly the same.(diminished can be used and thought of as dominant or leading tone, therefore, a function of the dominant, or pentultimate to I or i or other variations)
Dominant chords can be and often are subs through the tritone. (a glorious excercise, play dominant type scales or modes using the cycle of 4ths and 5ths. An average guitar players nightmare. A great excercise!) The irony is that there aren't a lot of tritones.
A major trick for every guitar player. Tritones have a distinctive look on the guitar. The idea is to make this stuff simple. The dominant and it's asundry variations make music, for the most part, palatable to the ear, regardless of experience.