If there's anywhere you enter username/password to access a site or account, then that info can be logged, and therefore yes, you are potentially compromising your security.
I have not followed past threads of yours to know why you have tried this. Unfortunately I have not tried any key loggers to know what they are capable of - but if ever a piece of software was capable of being modified for a secondary purpose, this would be it - for example, every 2 days or when a certain amount of text has been collected, the s/w could monitor how busy your internet connection is, and if you have a few things going on, could then "pump" all the data that had been collected off to some remote website... for who knows (?) to view.
Could inadvertently give away access to your bank (depending on how different the questions are each time you login).
As for windows live chat - sorry, have come across a number over the years, like AOL Instant Messenger, ICQ, MSN, Yahoo, and so on. You should be able to reinstall it without too many problems by visiting the website (Oh, was it "Windows Live" - part of Microsoft, you mean?).
Did you mean that nothing you typed in chat was logged ? It probably depends on how the key logger works - most of the time, your keystrokes cause characters to be put into a memory buffer, read by whichever application (browser / WP / calculator) happens to be in focus.
Some software may directly check the keyboard and turn off the buffer. Games for example. It would mean the keylogger would have no 'fresh' data to log, for the time nothing was going into the buffer. Some chat software might do this... perhaps to avoid being logged (!)