Yes, gazebos and sheds may require planning permission if they exceed certain sizes, or are located withing certain distances of the main house or boundaries.
From a planning point of view, they are treated as extensions.
But, there may be 'permitted development' rights, in which case no formal permission is required. Ask the planner if these are permitted development - may well be, if no other extensions have been built.
With regards to the shed, then if it has been there more than four years, then no enforcement action can be taken. So obviously, this was built before May 2003
With regards to the gazebo, I would be inclined to take the roof timbers off, invite the planner back (or send him some pictures) and ask him to confirm it is no longer a gazebo.
Then put them back on.
Then if another letter is sent, take them off again ... etc, etc
I am curious how a planner can ask if the guy had planning permission - surely he should know? Or should have said, "Yes thanks. Bye now, have a nice day"
Something to consider is that the LA should/can only take enforcement action if it is in the 'public good'. Action will cost them a lot of money, and time to deal with, and should only be done as a last resort if the 'development' is significantly affecting the amenity of neighbours, or the wider public.
I would stretch it out and see what develops. After all, a few months down the line after lots of letters and jobs-worth action, all that needs to be done at the last minute, is to take the timbers off so that it is no longer a gazebo.
Then put them back on again after a while . lol