Air Conditioner Iced
The icing is most likely really due to low refrigerant. I had similar problems with a slow leak in my heat pump, and limped by for a few years by having it recharged once a year since it was a slow leak. The bill usually fell between $150-300, gradually increasing every year since the leak was slowly progressing. I finally replaced the whole unit last year for $4500, since the refill schedule was accelerating down to 7 months in between rather than 12. (It was a 14 year old system, so that contributed a great deal to replacing vs repairing in my particular situation.)

Nitrogen is a common method for detecting leaks, so your tech wasn't doing something crazy. The new coil they suggested installing is probably the one that's leaking. You could ask to see where it's leaking, since they should have pinpointed the location with the nitrogen test. As for the blower board, they could be taking a shotgun approach to the repair and replacing parts they think could be related, but I can't say for certain, since they've seen it firsthand and I haven't (I'm also not a certified HVAC technician!). The blower could be a contributor if it's not running, since reduced airflow can contribute to icing as well. Try setting your unit on "blower only" mode, and if you get good airflow, that's probably an unnecessary replacement part. Or it would at least warrant further questioning to the repair company if it's really necessary.

I'm not sure what the wire is for without seeing a wiring diagram unfortunately. The fan screws should have no effect on icing, but it's a good idea to replace them. Remove the last one and take it to the hardware store to find a match. (Make sure you flip the breaker off first, since you don't want the unit running without it's fan attached, or spinning up while you're working on it!)

If you really think they're trying to pull one over on you, you could always get a second opinion, but you'll probably be on the hook for another $100+ basic service call fee from the next company too.