Not that any of this matters from a planning perspective, however what has your relationship been like with the neighbour? Do you generally get on or have their been challenges before? I note the boundary fence is quite high and as others have touched on, the wood store is slightly higher again. Was anything mentioned by the neighbour when both of these were erected, even in a half joking manner?
I think it's pretty safe to say your neighbours have probably been the instigator in the second planning/enforcement officer, however as you've nothing to worry about in terms of the garage and gates I'd try not to worry about that. Easy to say I know!
As for the extension, I'll play devils advocate. Whenever someone builds such an extension i.e. close to the neighbours property and any ground floor windows e.g. kitchen, it can be understandable that not every neighbour will be happy about it. That's human nature. However if your proposal ticks all the boxes required by local planners and they approve, you're 100% within your rights to proceed. The last house I lived in was semi-detached with kitchens adjoining. I have to admit, if my neighbour had built an extension from their kitchen, I wouldn't have been too happy due to reducing the view etc. However if said extension was approved, there's not much I'd have been able to do about it.
In short, if you're complying with the rules, you have the (legal) right to proceed.