is it possible to detect a boot-up via live CD (eg. Ubuntu, Debian, Tails ...) that happened in the past? If yes, where can this information be obtained?
I suspect my PC has been booted through a live CD, and got files manipulated,deleted ...
In general, it's possible for a live CD to boot a computer without leaving any traces. In some cases, this is an important feature of them - forensic imaging discs, for example, could change evidence if they wrote anything to the internal drives (and even then, professional forensic investigators usually copy using a hardware write-blocker if at all possible).
Therefore in this case, your best bet wouldn't be to look for evidence of booting from a live CD, but to look for evidence of changes to the filesystem which seem unexpected - files modified on a date that you were not using the computer (e.g. weekends, for an office PC), or at a time that the computer should have been off. You could also run an undeleter, and see if anything you don't expect to have been deleted shows up.