'At' the other side of the road is not correct?
Logically, it should be. Sadly, it isn't. However, I daresay it would be understood.
Road = estrada.
Street = rua.
What is the difference between "do lado" and "no lado"? When do I use which?
For me, There's no difference.
A minha casa está no outro lado da rua.
A minha casa está do outro lado da rua.
"é" and "fica" sound better to me here =)
why does duolingo offer "way" as a possible translation of "rua", but 'On the other side of the way' is wrong? Is "way" too general?
Yes, "way" doesn't sound right here.
Yep. It means only a passage, and it is inaccurate. The portuguese word for it is "um 'caminho'"