Whew! That's a lot!
First: Explaining all the "why"s is much easier if you can do some reading on our sister site, poolsolutions.com

All professional pool installers and repair guys have their own preferences, usually based on what they know.

Inline vs floater: Both ASSUME you will be chlorinating solely with 3" Tri-Chlor tablets If you learn our B-B-B system you'll find out the tabs can be great, or seriously problematic. They add stabilizer and are VERY acidic. Personally, I stick to Chlorine Bleach or its stronger form, Liquid Chlorine. It's cheaper than tablets and more controllable. When I use the Tri-chlor tabs, I use a floater--they are about $10 versus far, FAR more for inline chlorinators and don't need to be installed. I also use a Salt Water Chlorine Generator so my call use tri-chlor at all is very rare. But the nonsense that floaters don't work is....NONSENSE! He just wants to sell you an inline + installation.

EVERY type of filter if sized and maintained correctly can give fine results. The second guy (again) is trying to sell you a sand filter + installation. If the cartridges are good, then just use that. (Personal bias--I have a sand filter and wouldn't buy anything else---but if my pool came with a cartridge filter in good shape, I'd use that).

I don't know what your "shock" treatment is. It could be all kinds of things, but mostly they are either Di-Chlor powder or Cal-Hypo powder. Either, dissolved in water should kill the algae in the filter cartridges. But so should hosing them down. Personally, again, I'd use bleach or Liquid Chlorine.

Sounds like the first guy is trying to save you money and the second is trying to separate you from it!