Are there any ways to figure out if the copper foil and mesh I have are pure, as-close-to 100% copper as possible (and as advertised)?
As a home tool, but not necessarily a definitive solution, I would suggest the dissolution of your copper metal/alloy in a galvanic cell where the Cu is the anode and as a noble cathode, use a source of graphite.
This approach for the dissolution has been favorably argued in commercial patents in a 1999 U.S. Patent 6,692,632. To quote in parts:
The method is preferably used in synthesis, but can also be useful in the destruction of pollutants etc. In certain embodiments of this nature, the use of external current applied to the system is not necessary, and a simple galvanic cell can be set up...
On copper particularly, for the home chemist, I have demonstrated a so-called Bleach Battery approach as an electrosynthesis path discussed in 'Subject: Experimental Variants of Bleach Battery to Obtain Metal Oxides (like CuO). However, I would replace the cathode, as in the current case as Nickel could be present as an impurity also in your copper metal sample, with very noble graphite in its path to CuO and other metal oxides that may become evident (or be chemically identified).
As the copper can be alloyed with metals like Zinc, Tin, Aluminum, Nickel, Silicon,..., with the first three creating respective oxides from the dissolving a small piece of the copper metal/alloy, potentially revealing distinct metal oxides as, for example, CuO is black, ZnO is white, SnO is blue-black (as produced by heating the tin(II) oxide hydrate, SnO·xH2O where x<1), and Al2O3 varies from white to a clear transparent gel. The metals Nickel and Silicon are expected in their metallic state.
I say, convert a bit of it into metal oxides and proceed from there!
+If you are in the US ( and many other places)you have copper 101 or 102 ( CDA = Copper Development Association designations), with 99.95 % minimum Cu. CDA says this Cu is what is supplied under the following ASTM Material Specifications- B48, B132 ,B152 ,B187 , B272 , B432 ,and F68 ; all flat Cu products. The same Cu is supplied in other shapes in other ASTM specs. Do you know the ASTM designation of your material ? What is your application where 99.95% + Cu content may not be sufficient ?