@Nonsense:
I am wondering if activating Snort is really necessary in my home environment (I have set no ports open)?
My answer is not hardware related, but:
I assume you mean all ports from Internet is closed, but all ports outward is open.
If/when your computer is compromised by malicious code, I can bet that it happens while surfing the web and a webpage will take advantage of a vulnerability on your system. (PDF, flash, browser, java or similar). The malicious code might have a call back function so it connects to a server that either upload new mailicious code, adware, or are able to control your machine. This connection will be done in intervals and will never be initiated from they outside, but from your computer. So having all ports blocked will not help you.
Snort, might be able to detect this traffic, but that depends if you are willing to use time to tune Snort and actually use time to investigate the alerts. If not, you will not have any advantage of using Snort.
You would probably use less time by having your system patched all the time and run scans of your systems now and then. Minimizing the risk of actually being compromised by surfing the web.
While I'm at it, here are something I would recommend (even if you didn't ask):
Personal Software Inspector (PSI) - Free for personal use. Scans all programs installed and report of outdated programs. Really useful
Qualys Browser check - Checks if your browser (Chrome, FF and IE), plugins, FW, AV, WU are vulnerable or outdated.
Firefox plugin checker