My two cents.
Write a letter assuming responsibility ONLY for the dirt and dirty windows (you don't want them to make it bigger), state clearly that you are going to hire a professional (not your jerk friend) to do the clean up and at there convenience. Leave it the f*ck at that and enjoy your house, don't let them intimidate you to stop doing house chores. In regards to the Dog, remember it's not the Dogs fault so chill about that, let him bark and call the Cops or garner evidence if the Dog is causing you to loose value of your residence.
They are n*****ts and I have one asswipe next door too. The damage is done, it was a mistake, you own responsibility and you want to fix it. Don't get caught into there hang ups.
Life is too short to sweat the little things.
I can only relate to my experience. I have an asswipe next door neighbor (he reads this group BTW). When we first moved in (our first house coupled with watching HGTV) my Wife and I have not stopped working on our property doing improvements. Another neighbour up the street told us that our next door neighbour will not like us for trying to make our property look nicer than his. So maybe the dirt you said was blown over (small you said) is just a red herring, I just don't see any one getting into hypochondriac fit over another neighbor trying to do improvements.
Good luck Sir and as someone already said, good fences makes for good neighbors.