TDS meters are really conductivity meters that are usually calibrated for sodium chloride. If you have something other than sodium chloride you will get a conductivity reading but the TDS won't necessarily be the same as if a laboratory test is done.
The laboratory test is based on evaporation of a certain abount of water and weighing the resulting solids.
No. The critical part of the meter is the electrode that presents a controlled area to the electrolyte, often includes temperature compensation, and converts the whole thing to a measure of conductivity.
The one I have doesn't have a name on it. I bought it from a Well Drilling Magazine I get monthly, but can't remember which one. If you really need one, I'll start looking for the ad.