I know there are plenty of words that use the -st ending: wouldst, whilst, unbeknownst, etc. but I'm not really sure what it means to add an -st suffix to a word. What does it mean to add the suffix? How can I tell what words can take the -st suffix? Are there any modern words that can take the suffix and not have people look at me in a strange way?
Long long ago in a galaxy far far away...uh, well, not really.
English used to have a more complex grammar than it does presently. It is a Germanic tongue and so retains a touch of German in old, not so much used, forms.
The -st you refer to are from the old second person singular. Wouldst:
As to whilst, it is also a contraction of "whilest thou", it is an archaic formulation which is still used, although more in Britain than the US. See Wikipedia on Whilst. and its place can be taken by "although". It is also common these days to use "while" in its place. "Whilst we watch Rugby, the Americans watch Football." becomes "While we watch Rugby, the Americans watch Football." Put "Although" in that sentence instead and the meaning remains the same.
In short it is an archaic form, and to use it instead of that which is more modern might be seen to be pretentious, or poetic.
On the other hand, while "whilst" and "wouldst" are uncommmon, I have heard "unbeknownst" used much more commonly: "Unbeknownst to me, she was married." This would likely not sound odd at all to me, just perhaps a touch formal.
Another one that is completely disused is "durst", which is the second person singular of "to dare".
For unbeknown vs. unbeknownst: http://fandom-grammar.livejournal.com/39346.html which would probably explain the others also. "Whilst" I've heard British people use working with them, but everything they say sounds funny and lovely!