A modified thinset, when installed properly, is stronger than a dryset (unmodified), so the use of it in a shower is fine.
Re the mastic on the walls; IF the tile size is within range of the mastic chosen (bigger tile cannot be successfully installed with mastic), then it is okay, but it actually costs more than thinset in materials. The advantage is you don't spend time mixing it, and then cleaning up your bucket and paddle or throwing away some that begins to set before you can use it.
I am concerned about the floor install. What you describe is NOT an approved, tested installation procedure. It is common, but it has never been successfully tested in a long-term install. It is commonly referred to as a Jersey Mud job. It does not show up in the Tile Council of North America (TCNA) handbook, because if fails the Robinson field test, the industry standard for suitability of installation methods.
There is certainly more than one approved method to install floor tile, and the only one I'm aware of that uses lath is one by Mapai (you can look it up on their website, but it uses their plastic lath material and a very specific mortar) or a full reinforced mudbed, which over a wooden subfloor must be at least 1.5" thick. There is no other tested/approved method of tile setting using lath that I'm aware of. Have the guy show you an approved method from the TCNA handbook that matches his plan...he can't! Probably doesn't know what it is, but it is the golden industry standard.
The more common methods in use are a cement board (cbu - like Hardie Board), or a decoupling membrane (like Ditra). Both of those, if installed per the manufacturer's instructions, are reliable and work well.
You never did answer what the subflooring and joist structure consists of. If that is not up to industry standards, nothing you put on top of it has a reliable chance of survival, regardless of the method. Every approved method starts with a suitable substructure.