First off, you will need to provide more information in your question to help this audience narrow down and assist you. As you have proposed it, your question is like asking - my house might have been broken into, how can I tell while I am on vacation in africa.
Secondly, having installed linux systems (fedora being one of them) on a variety of hardware configurations, you can sometimes get this behavior just from the system. Lately, for instance on my Ubuntu system, I have been finding for some strange reason, when I connect USB drives (clean ones I know for a fact) to my machine, my wireless mouse starts doing a very similar behavior - it becomes erratic. So what you are witnessing could simply be a kernel change that is flaky.
Finally, the advice provide above is good, however some pointers and thoughts:
If a rootkit has been installed on your machine, understand any utility (including netstat) could be subverted, so you can not test what is being reported. If you are wanting to listen to network traffic coming from/to your machine you will need to do it from an independent machine. You can do this very easily by grabbing a Snorby appliance and setting it up on another machine (its a VM) on your subnet and listening to your traffic. (https://snorby.org/)
Boot your machine with a clean fedora livecd and gather md5 hashes of all the key binaries. Then go set up a clean instance in a VM and bring it to the same run level as the one yours is and capture the same hash base. Do a diff.
Run through the SAN checklist provided.
Google around and see if other users of your fedora version/level are having similar issues with the mouse config you have. I would do this FIRST before anything else.
Just a few of the quick things you can do. You can be as diligent as you want, and take it to whatever level you want. Its up to you and the time you have.