Can I hook up my roku to a different house and watch my tv still?
Yes, the Roku devices are meant to be portable. Just remember to bring the Roku remote with you or you are going to find it hard to connect to the new network at the new house.
Just plug the Roku device into the HDMI of the new location's TV and connect to the network.
All your content will still be available to you at this new location.
Yes. You'll just need to change Wi-Fi credentials, unless you are using a hotspot.
Actually, not always.
If the two houses are in different countries (you didn't say same country) then some channels won't work (country specific ones, not things like Netflix). Better to have dedicated device and account per country. Similar issue if you take Roku with you on international trips.
Also, not quite sure what will happen if you have different cable providers in each house and have some channels which are linked to cable subscription.