"We live here, beside the wide river."
Translation:Itt lakunk, a széles folyó mellett.
Theoretically speaking, yes. But as a general rule, put the itt/ott before the verb, it is the safe solution. In the case of the original sentence, there is not such thing as "lakunk itt" in normal indicative mode. You can ask "lakunk itt?" which translates to "do we live here" but the true meaning is a joyous, lively exclamatory question when you find your perfect dream house and ask your family/SO/friends/whatever if they want to actually live there. When you just normally ask "do we live here?" you use "itt lakunk?".
The example is more clear on second person. "Itt laktok?" is when you point at a house and inquire if your friends live there, like when you are looking for the right house in the darkness. "Laktok itt?" is actually inviting your friends to move in with you.
"Van itt madár" is a variation of "itt van madár". The change of word order here essentially emphasises ther very existance of birds, like "birds do live here".
The same applies for questions. "Itt van madár?" and "van itt madár?" are the same, only the emphasis is different. The first focuses on the location and the second on the existance.