Is it a bad idea to paint the underside of a deck

For the main deck, I would recommend a floor sander, similar to what's used for finishing hardwood floors. Put a 100-120-grit belt on it and take it back and forth along the deck in the same direction as the planks until you see bare wood. Once you have bare wood, I would apply whatever stain or sealant you want very quickly thereafter, as you have just removed a large portion of both the artificial and natural weather and bug resistance the wood had.

A handheld orbital or belt sander will get most of the stuff off the vertical components (handrails and pickets). You'll need a "corner cat" or other detail sander, or some elbow grease, to get the paint out of crevices and detail work.

You will not be able to avoid taking off some wood along with the paint. Paint (especially a product designed for this kind of outdoor application) will seep into the grain of the wood, and you will be removing wood before you see the color disappear and bare wood show through.

By the way, you will still need to maintain the deck after staining and sealing. The stain will last for quite a while, but you'll need to re-apply the water seal at least once a year, once every 6 months if you get a lot of rain.

Painting a pressure treated wood deck is always tricky. Pressure treated wood does not seem to hold paint well, even with a good primer. Stain is usually a better alternative, but since your deck already has paint, it would have to be stripped completely before using stain.

In order for the paint to stick fairly well, the surface needs to be as clean and dry as possible. Looking at your pictures, I see a fair amount of algae in the grain of the wood. This is a very normal condition. You should however, use the pressure washer and bleach/water/TSP shrub down method to clean it up. I'd wait until April or May when temps are up in the 60's to 70's to help dry the deck and open the grain.

Select a high grip primer rather than a stain blocking type. Many manufactures have come out with new formulas that are designed to bond better. When you apply the primer, don't rush. Simply rolling on primer too quickly is not good. The primer needs to be brushed into the grain as deeply as possible. Let it dry completely. Now you can apply your deck paint as normal.