can my landlord see what i do on my personal laptop?

It's possible, but requires more work and expertise than your landlord is likely to invest.

I'd be more worried about that spy camera in the bedroom lamp.

Does he have access to your computer? Does he have his own computer with the same connection? If not, I would think he can't. If you were using wireless internet on his network and he had his own computer, he could be able to see what you're doing or hack into your system, but it's unlikely he has that kind of knowledge anyway. It's not something your average person would know how to do. I'm sure you're fine.

Now, if you were do something illegal, he would be the one at risk because the internet is in his name. Don't look at any child porn and you shouldn't have any problems.

Extremley doubtful especialy if not wireless. If he has physical access to laptop then maybe but if aint I doubt it. Plus without your permission its illegal

no he cant. you see what ever you do on your personal laptop is saved on your laptop (Internet and everything) not on the Internet.

No he can't see anything unless he has your laptop/Pc