"No! Marcus sleeps at home."
Translation:Minime! Marcus domi dormit.
'Domi' is a special case. By itself it means 'at home'. With just about any other word you would need a preposition.
(Nota bene, this is also true of the word 'home' in English: you can say 'I am home', with no preposition, but you cannot say 'I am city' or 'I am airport' or 'I am a public toilet'.)
'In domi' would be like saying 'in at home', which is of course wrong.
Technically, the Romans were writing before modern notions of capital usage was invented. So inscriptions were in all caps
whereas handwriting was similar to all lower case
(There are samples of this from Vindolanda and other Roman forts, as well as curses from all over the Empire.) Modern lower case came from mediaeval minuscule script. So if you're imitating Roman handwriting, you can go with all lowercase. But modern practice would be standard capitalisation. For Modern Latin, lol.