Building works will never make you popular. Mess, noisy machinery, huge deliveries, vans blocking the road... Try doing it all when you live on a narrow 25% hill*. For matters like the gate, just get it done, your neighbour has no reasonable cause for complaint. And remind her that your house looking nice makes her house worth more.
Our neighbour is mostly fine (we have a similar setup with a wall only accessible via her property). Our only gripe are a couple of self-set sycamores and a hazel that grow in the gap which really need to come out (they'll just overgrow our house, the roots are a level below and will eventually damage her property rather than ours, everything is terraced owing the hill). She keeps agreeing to get it done, but this is year 2 of her conspicuously not getting it done. I figure I'll just offer to pay. Mildly annoying as it would have been easier before the leaves came.
As for the dirty car, 'fuck off' is I think is the only reasonable response. Our car is parked under cover and rarely used and it's still covered with dust within two weeks. That might be our neighbour's wood stove, she burns anything too (can't complain, I pass plenty of it over the fence). I'm not bothered, cars get dirty.
*for bonus points, have you removals pantechnicon block the hill at about 5.30pm on a weekday and create your own traffic jam. Most people were OK about it, the movers were helping people squeeze their cars by, but there was one idiot – BMW natch – who came banging on the door demanding that I have them move the lorry because they were 'blocking his right of way'. He was ready for a right barney. Bad fight to pick mind, considering I standing there with no fewer than eight blokes who lift furniture for a living.