I don't think there's anything to stop an owner inspecting common property unless there are restrictions mentioned in the by-laws (times of access to a swimming pool, for instance). But accessing their rented property is a whole different issue.
Who issued the letter and on what basis was it issued? Is there a by-law? Is there a fire safety issue? Is the stuff a visual intrusion? Without that, anyone who removes another person's property without proper authority could find themselves in all sorts of trouble. I'd be photographing it as evidence for the claim against the Owners Corporation when it goes missing.
The arbitrary issuing of “naughty” notes by members of the EC has little legal standing. For any such warning to have any effect, a “Notice To Comply” has to be issued following a properly constituted Executive Committee meeting (unless a Strata manager has specifically been delegated to carry out these EC functions).
The phrase “cannot unreasonably be withheld” with regard to permission for pets is marvelously loose. But if the dog is neither dangerous, nor a nuisance nor makes a mess of common property, its size is irrelevant. Great Danes are amongst the best pets you can have in an apartment because they are lazy and non-territorial.
If there is no actual reason for refusing the dog then, by extension, refusal must be unreasonable.
Going by other discussions elsewhere on the Forum, you probably can't change the number of members of an EC at an EGM but if you can raise a 75 percent vote in favour you can remove individual members (or the whole committee) and replace them.
Sounds like it's time for you to sit down with The Brothers and explain that times have moved on and strata blocks can't be run like institutions any more.