I think you've gotten some really good advice on handleing the dynamic taking place in your office, so I'm only going to offer a bit of advice on the Password/log in scenario.  You might not like what I'm about to say, (but let's face it I'm infamous for being direct).

the information contained in that computer belongs to your company.  I agree with CG above who stated that you need to abide by policy,but even if you've been told not to give out the password, rest assured that with what you've told us about your colleague she will apply for and find a waiver to that policy using the excuse that you're going on vacation.

There should never ever be anything on a company computer that is not 100% transparent to the rest of the organization.  If you wouldn't post it on the bulletin board in the break room, then it shouldn't be there.  E-mails, calendars etc should be a matter of record for everyone within an organization.  If you're netsurfing on your lunch  break, fine and dandy as long as your company policy allows that.

go immediately to your employer and mention that your workmate has asked for the computer password and ask for instruction as to whether you should provide it. And how you should present a possible negative answer to this colleague.  If bossie tells you Not to give her the passwords, your immediate response should be to get his advice on what you should tell your coworker and stick to it exactly without deviation.

Even if your employer tells you NOT to provide the passwords, this employee sounds like she has her own agenda and you can pretty much assume that some "emergency" will arise which means she needs access and will get it.  She's connected, she's been there longer and you my dear are a threat, whether you mean to be or not.

Take an hour before you go and look at the files contained there and make sure there's nothing there that shouldn't be, and in the future resolve to never have to ask yourself if there's information on a company machine that couldnt be seen by the CEO on a bad day and be misread. Cause IF there's anything on there that CAN be misread by others, be very sure it will be looked over by your workmate at some point during your absence.  When people identify others in the workplace as a threat to their position or to their level of "imtimacy" with the bossie, they don't always behave in the same way that they would in other professional situations.  

If you and bossie have specific files that you don't think this person should see, then back them up to bossies machine and delete them from your machine, restoring them when you return and in the future, store information which is collective for you bossie and which you BOTH feel is sensitive on his machine rather than yours. Because you both (you and the other employee) both have access to bossie's email, I have to consider that it's more personal information you're worried about your colleague seeing, rather than business information.  IF (And I am sorry if I've made a bad supposition here! ) IF That's the case, remove the offending information and make it a PPP (Personal Professional Policy) that you never EVER put anything back into a work machine that's not 100% to do with objective work and workflow process.  Doing so means you can't send an email to a colleague which says,,,, "GRRR! Xyz is driving me CRAZY today".  And Trust me that's an excellent policy to have.

My tuppence,,,,
Andrea