Is there a guaranteed way to:
Know there is a Remote Access Trojan in my PC?
Remove it completely and successfully from my PC?
Note: Assume that the hacker doesn't leave any hint of their activity (like moving the cursor).
Proving a negative is basically impossible. It's like saying "How can I prove that nobody owns a key to my apartment?" - you simply can't.
However, you can detect certain remote access kits through a variety of techniques. Keep in mind that these techniques require some level of expertise. People often say "Look at your network traffic", and then they go buy a tap, have a look in Wireshark and see lots of network traffic to various domains and IP addresses they cannot explain and then end up believing they must have been compromised. I personally label this the "Wireshark Syndrome". However, the underlying idea is correct - look for suspicious traffic and investigate further. This requires you to be able to tell suspicious traffic apart from benign traffic, which is very difficult.
Another possibility is to use a ready-made tool for RAT detection. These tools look for traces of know RATs and give you a warning should it find some. The upside of these is that they are a lot more user-friendly, but with the downside that they also cannot 100% detect every RAT out there. "I couldn't find anything" does not mean "There is nothing", after all.
How can it be removed from my PC?
The question Help! My home PC has been infected by a virus! What do I do now? goes into detail on how to deal with a compromised home machine.
tl;dr: Nuke it from orbit!
How can I make sure my PC is safe?
As mentioned above, you can't. What you can is follow best practices, that are repeated over and over again by security experts:
Of course, this list isn't exhaustive, but if you follow these guidelines, you will likely not be "low-hanging fruit", and that already gets you out of a lot of trouble.
