1. don't always think on heating/burning that your compounds should loose mass. you might be forming some sort of oxide or something?? My suggestion is that you go back and redo this bit and see if you get the same thing.
2. your solubility can be worked out by simply dissolving small amounts of your salt until no more can be dissolved!! just keep track of each addition you make, and make sure that you keep your solution at about 20o
C so it matches standard lab conditions which most references will be at. final result will be in grams per litre (g/L)
4. Spectrum/spec-20 see the connection? I assume this is just your run of the mill spectrophotometer and yes this will be what gives you the absorption wavelengths for you compound. This will just be a matter of dissolving your compound in the right solvent and running a spectrum with the machine and just picking out your major peaks.
5. Ok well this is where solubility rules come in. most first year chem textbooks will have a list of them but this website is also a good source:http://www3.baylor.edu/~Kenneth_Busch/Solubility_Rules.html
Basically you have to go through a process of adding different solutions to a solution of your unknown and see if aything precipitates out. Each time something precipitates out it gives you a clue as to what it is.
Also have a look at this procedure:http://www.uncp.edu/home/mcclurem/courses/chm226/Salt_Unknown.pdf
Looks like a fair bit like what you've been asked to do.