I received three bags of stamps still stuck to paper (I was only meant to get one so I'm happy) and an envelope full of dog and cat stamps.
Kayleigh received a letter inviting her to make an appointment to get a pap smear. 
Modern stamps will often peel right off if you want to. To me it would depend upon the paper they are applied to before I peel. Then, you still have to figure out what to do with the "glue" on the backs. There are various sizes of plastic mounting containers available in strips which you cut to length which help.
Acidic papers can affect the inks and damage the stamps. There are little "litmus" pens you can find that identify pH and warn of sulphur or phosphor bearing compounds which could be detrimental to the preservation of your collection. One small line (specialized color changing "ink" pen) on the back of any paper will help to identify many harmful agents which could hinder archiving some papers.
Obviously, The Smithsonian does not use them (they have better ways), but an envelope that has crossed continents, oceans, even vast amounts of time can be quickly identified as to its make up.
Older stamps used water soluble adhesives. Hence the whole "lick a stamp" thing, which has always put me off some.
Older stamps can be recovered from envelopes by soaking them for a few minutes in room temperature water, laying them on a cloth to dry, then pressing them in your stamp albums to relieve any curling. Just be sure they air dried completely before you try to store them.
You probably knew that, but now several others do as well.
Sorry, but typing one handed is still a bitch with a whip!
