I am shocked you have not had problems with your X5. My 2004 4.4i hated cold weather. It would "earthquake" on start up and the dash would light up like a christmas tree. The door handles broke all the time in the cold. The rear washer nozzle would freeze and then spray continuously until you pulled the fuse, and the wiper arm would break if ice was on it. The Xdrive hated the snow and ice.
I have had to do some repairs, but nothing I would consider out of the ordinary for a vehicle with 178000 miles.
Yes I also had to replace the door handles. They are like $40 on Ebay brand new, but the quality varies a lot.
I also had to replace the radiator,the brakes, battery, tires, windshield and the front control arm bushings. But keep in mind this over a 3+ years period.
The shaking sounds like a bad coil. One common problem with the 4.4i is the valve cover leaks and the oil end up around the spark plugs and inside the coil, causing problems like you describe.
The reservoir for the washer fluid is in the front and it has two pumps, one for the front and one for the back. They are super easy and cheap to replace.
You probably just had a bad connection there.
I drove my X5 between Boston and Quebec all the time - I am actually going back tomorrow and it's about -15C at the moment.
Don't get my wrong, it's not a car for everyone. You need to be a bit of a mechanic for sure to stay on top of it and don't break the bank.
I do think that the X5 is better than most people are lead to believe.