Though it may seem odd, it is possible the plumber knows what he's talking about. If this is a Polybutylene pipe, it could indeed be brittle and fail if it's jostled during the installation of the filter.
It's impossible to tell from the picture what type of pipe that is, but based on the plumbers concern I'd guess it is Polybutylene.
Polybutylene
Polybutylene was used in pipes between about 1980-1994/5, and was commonly used as supply line in the 80's. It was found (in some cases) to react with oxidants in the water supply, which caused the pipe to become brittle and fail. To my knowledge, this type of piping is no longer used. Though if the house was built; or there was supply line work done, during the 80's, it's likely this is Polybutylene pipe.
High Density Polyethylene
Don't confuse Polybutylene pipe with Polyethylene pipe. High-density polyethylene is commonly used as water supply line today, and is stable and safe to use.
Cross-linked polyethylene
Cross-linked polyethylene (PEX) is not typically used as an underground supply line, as it can sometimes react with minerals in the soil. Where PEX is used underground, it's typically encased in sand (or other non-reactive materials) to prevent negative reactions. Because of this it's often more expensive to bury PEX, so other materials are used instead.
How can I tell what it is?
The only way to be sure what type of pipe this is, is to look for markings on the pipe itself. Polybutylene will be marked PB####, Polyethylene will be marked PE####, Cross-linked polyethylene will be marked PEX-x (where x = A, B, or C).
You have asked a lot of questions at once here. Let's start by saying that it is very unlikely that you would be able to determine the exact location of the leak from the top, unless it is a simple problem with the sealing around the shower mixing valve handle escusson trim or an obviously loose drain flange. If the problem is in fact under the floor, then you will have to remove part of the ceiling to gain access to the drain pipes for inspection and repair. It is not uncommon for joint to crack or come undone. I wouldn't consider removing the shower floor until it is inspected from below. Depending on how comfortable you are digging into a semi difficult project like this will determine if you want to hire a pro or tackle it yourself. It will most likely involve removing a section of the ceiling, repairing some PVC pipe or making a new joint with pipe and couplers, primer, pvc cement etc., then repairing the hole in your ceiling.
Best Answer
Yes it's normal and what you want. It's called a P-trap and the water sitting in there seals out the sewer gasses.
My ASCII P-trap:
The asterisks would be the water that forms the stopper.
In an empty home or in drains that are infrequently used (such as a basement) you these traps can eventually dry out from evaporation and let sewer gasses seep in.
In your case, if you aren't going to use the shower for months on end, I suppose you could add a drop of bleach to the drain. If the shower is used semi-regularly, there should be no issues at all.