When the whole sentence is read, the stress in дома is on the "O", but when I hover over the word, it's on "A". Which one's right?
This is the only way. «В до́ме» would mean 'in the house, in a house'.
дома here is genitive case, correct? Object of understood preposition "at"?
I've also noticed the use of accents over vowels, indicating stress on that syllable. Is that just an aid here to beginners, or is it something some people do in the language itself, in ordinary use?
It should be «читаю», not «четаю».
This word order shifts accents:
- «Я чита́ю до́ма» is a sentence about where you usually read. The new information: your preferred place for reading is your home.
- «Я до́ма чита́ю» is a sentence about what you do at home. The new information: your preferred activity at home is reading.
Those sentences would be used in different contexts, when discussing different things. I think your variant is more likely to be translated as ‘At home, I read‘ into English, but I’m not a native speaker so I can’t say for sure.