Many people seem to be afraid of mixing stable with testing, but frankly, testing is fairly stable in its own right, and with proper preferences and solution checking, you can avoid the "stability drift" that puts your core packages on the unstable path.
"Testing is fairly stable??", you ask. Yes. In order for a package to migrate from unstable to testing, it has to have zero open bugs for 10 consecutive days. Chances are that, especially for the more popular packages, somebody is going to submit a bug report for an unstable version if something is wrong.
Even if you don't want to mix the environments, it's still nice to have the option there in case you run into something that requires a newer version than what is in stable.
Here's what I recommend for setting this up:
First, create the following files in /etc/apt/preferences.d:
stable.pref:
# 500 <= P < 990: causes a version to be installed unless there is a
# version available belonging to the target release or the installed
# version is more recent
Package: *
Pin: release a=stable
Pin-Priority: 900
testing.pref:
# 100 <= P < 500: causes a version to be installed unless there is a
# version available belonging to some other distribution or the installed
# version is more recent
Package: *
Pin: release a=testing
Pin-Priority: 400
unstable.pref:
# 0 < P < 100: causes a version to be installed only if there is no
# installed version of the package
Package: *
Pin: release a=unstable
Pin-Priority: 50
experimental.pref:
# 0 < P < 100: causes a version to be installed only if there is no
# installed version of the package
Package: *
Pin: release a=experimental
Pin-Priority: 1
(Don't be afraid of the unstable/experimental stuff here. The priorities are low enough that it's never going to automatically install any of that stuff. Even the testing branch will behave, as it's only going to install the packages you want to be in testing.)
Now, creating a matching set for /etc/apt/sources.list.d:
stable.list: Copy from your original /etc/apt/sources.list. Rename the old file to something like sources.list.orig.
testing.list: Same as stable.list, except with testing.
unstable.list: Same as stable.list, except with unstable, and remove the security lists.
experimental.list: Same as unstable.list, except with experimental.
You can also add a oldstable in sources.lists.d and preferences.d (use a priority of 1), though this moniker will tend to expire and disappear before the next stable cycle. In cases like that, you can use http://archive.debian.org/debian/ and "hardcode" the Debian version (etch, lenny, etc.).
To install the testing version of a package, simply use aptitude install lib-foobar-package/testing, or just jump into aptitude's GUI and select the version inside of the package details (hit enter on the package you're looking at).
If you get complaints of package conflicts, look at the solutions first. In most cases, the first one is going to be "don't install this version". Learn to use the per-package accept/reject resolver choices. For example, if you're installing foobar-package/testing, and the first solution is "don't install foobar-package/testing", then mark that choice as rejected, and the other solutions will never veer to that path again. In cases like these, you'll probably have to install a few other testing packages.
If it's getting too hairy (like it's trying to upgrade libc or the kernel or some other huge core system), then you can either reject those upgrade paths or just back out of the initial upgrade altogether. Remember that it's only going to upgrade stuff to testing/unstable if you allow it to.
EDIT: Fixed some priority pins, and updated the list.
Best Answer
I do not think it is possible for anyone here to properly figure out what is wrong with your system, unless we have direct access.
You can assume it is a false positive and that is what most likely is the case. In my experience I have had the odd false positive as well and after some investigation I had satisfactory confirmed to myself that indeed it was a false positive. Some processes can be hidden from the ps command for perfectly legitimate reasons. And are often short-lived.
However if you want to be sure you could do something as described below. I understand it is a bit extreme and may not apply at all in your situation, but I can't be sure of that, that is a decision you need to make.
I personally take a lot of wisdom on what to do in such and other situations, from Securing Debian Manual, especially in your case check Chapter 11 - After the compromise
The general advise is to unplug a compromised system from the network first and foremost, then back up the system. Then perform more investigation. This should be a separate activity and doesn't need to get in the way of restoring the system. It'd be a good idea to try and find out how it was compromised.
In the meantime completely wipe the system, re-install the OS from a non suspect read only source (say a dvd) and restore from backup only the data which you can be certain of is not infected. If such data is contained in repositories, then just retrieve it from there instead of the backup.
Assume any files on the system and its backups are suspect and try as much as possible to restore it without copying files from the backup. Restore configuration files by just repeating what you did, copy pasting text etc.