Any system connected to the internet is going to get scanned, and often. If your system was behind a firewall which was set up to only allow outgoing connections then you would not be seeing these messages. This leads me to believe that you either have a direct internet connection or you are behind a firewall but it is set up to forward packets to your system.
The best thing you could do to protect your system would be to either put a firewall between your system and your internet connection, or if you already have one to configure it to only allow outgoing connections.
EDIT: Poster says his PC is directly connected to the internet
The main issue with your setup is that all operating systems have zero day vulnerabilities which could be used by an attacker to take over your system no matter what software firewall and antimalware software you are running. These are often integrated into automated software which scans the internet trying to infect as many systems as it can find, so even if you are extremely diligent about installing every single software patch you are vulnerable. The only way to reduce this risk would be to purchase a hardware firewall to protect your system.
Regarding your concerns regarding your employer it's unlikely that they've been able to install monitoring system behind your back, however monitoring software could have been installed as part of a wider software package, so if you were asked to install software on your system it is possible that it could be monitoring you. Also, java applets could hook into your camera, microphone, or browse your files. I'd say the likelihood of that being done is low if taken across all employers, however you have a better idea as you work for them. Plus they seem to have made statements to the effect that they are monitoring you, which could just be them joking but maybe not. If you are being monitored and you live in a place where there are employee laws that will protect you then it's a matter for law enforcement, if not then you need to take steps to protect yourself.
What I would do is either buy a second system and do employer work on that second system. You can physically disable any camera or microphone on it and as long as you keep it purely for work your employer will never see anything personal. If you cannot do that the other option would be to create a virtual system within your physical computer and use that for employer work. You can configure the virtual box to have absolutely no access to microphone or cameras on your physical system, and the virtualization will allow you to separate your personal and work data within one physical system. Just be sure to strictly limit the access from your work system to your personal system no matter what method you use.
Of course you could just tape over your mic and camera if you are worried about the physical surveillance aspect.