Fivey/Ainley, or any Doctor/Master if they don't work for the question.
Gonna try this even though my Fifth Doctor knowledge is limited.
Who cooks: The Master makes the meals, The Doctor makes desserts (they tried flipping it around one and the results were horrible).
Who does the laundry and other chores: The Doctor likes to keep the TARDIS tidy, while the Master is too busy with experiments (and, let’s be honest, schemes) to worry about the mess. If it were his TARDIS, he’d be much more tidy.
How many children do they have: Since Fivey traveled with Adric, Nyssa, and Tegan, I can see them having three children.
Who’s more dominant: The Master, no question. Which is a shame sometimes, because he gets really turned on when the Doctor tries to act dominant.
Favorite nonsexual activity: Something science-y maybe, like trying to rebuild Kamelion or “fix” part of the TARDIS. The Master keeps insisting his TARDIS is better and that they should be using it for traveling, but the Doctor is of course completely against leaving his TARDIS. So the Master just tries to add features from his TARDIS to the Doctor’s TARDIS.
Their favorite place to be together: The console room or the bedroom.
Any traditions: Pissing off the Time Lords or annoying the Rani.
Their “song”: Something instrumental, maybe the music of the spheres.
What they do for each other on holidays: The Master indulges the Doctor’s love of Christmas and lets him decorate the console room and their bedroom. The Doctor in turn lets the Master choose their holiday destination.
Where did they go for their honeymoon: The Eye of Orion.
Where did they first meet: The Academy (pretty sure that’s canon).
Any pets: Kamelion (lol) or a cat.
What do they fight over: The Doctor’s love of Earth, the Master’s schemes, the Doctor’s reluctance to trust the Master, and if he’s feeling especially angry, the Doctor will call out the Master for stealing Tremas’ body (he really like Tremas!).
Do they go on vacations, if so where: Yes, everywhere; the Master likes to visit worlds with more advanced scientific achievements than we have on Earth and that he can copy for his own uses, while the Doctor is torn between visiting Earth and keeping the Master away from Earth. When they do go to Earth they stick to the earlier centuries when there’s less technology for the Master to manipulate.