Playing Dr. Young's Monster Tuba is not as difficult as it looks. You only need three fingers on each hand, so Mickey Mouse (and most other cartoon characters) should be able to play this tuba. It is easiest to think of the fingers on your right hand as operating valves 1 (index finger), 2, and 3, while the fingers on the left hand operate valves 6 (index finger), 4, and 5. In a pinch you could play this instrument like a regular tuba (3 or 4 valve), sousaphone, euphonium or baritone, using the same fingering and the first 3 or 4 valves, but the intonation would suffer.
To get the tuba to play in tune, you should avoid valve combinations (even though some of them work) unless it is valve 6 and one other valve. Valve 3 is like the 1-2 combination, valve 4 is like the 1-3 combination and valve 5 is like the 2-3 combination. So you would play BBb with no valves, A is 2, Ab is 1, G is 3, Gb is 5, and F is 4 (when you are playing a low F).
Valve 6 is the switch valve, so if you can't play the E on the BBb side using valve 2 (because you are playing a low E), you can play it with just valve 6 (which is like the 1-2-3 or 4-2 combination). If you want to go lower, a half step at a time, you would play 6-2, 6-1, 6-3, 6-5, and 6-4. That would be Eb, D, Db, C, and B.
If you want to play real notes (down where the normal tuba's pedal tones would be) you would use the following fingering: Bb 1-5-6, A 1-4-6, Ab 4-5-6, G 1-4-5-6, Gb 3-4-5-6, F 1-3-4-5-6, E 1-2-3-4-5-6. After that, you would be using pedal tones on the EEE tuba, perhaps Eb 3-6, D 5-6, and Db 4-6.