Does Android suffer from the same "over a period of time" performance degradation that OSes like Windows do? To clarify, yes, I run a custom ROM on my phone. I thought to myself "I'm going to flash this and keep it stable because I need to depend on it for work".

Almost a full month later, I'm finding that the interface is lagging, apps don't spring open with the same swiftness that they used to, and in general, I'm finding I need to reboot more often. Nothing is critically wrong, logcats don't seem to show any malfunctioning, it's just.....sluggish. 

Flashing the ROM over again is starting to feel like Windows; I reinstall fairly often to keep it running smooth, but if it's something I'm doing wrong, I'm entirely open to suggestions. As I grow older and less obsessed with flashing anything I see on XDA, I want to keep something stable and dependable.
The effect you are describing is not caused by the OS. It's caused by more and more background processes that accumulate over time and maybe more apps registering to system hooks that need to be evaluated when certain events occur.

Every OS that supports this will be vulnerable. My Android has run for the past 1 1/2 years without any performance degradation. Always question and confront app developers if they run a service in the background for a long time (without any need/use). Interval-like started background services that check/pull for some information and terminate after that are fine. Google Play Music is was1 a perfect example: The App sets an Android Alarm (that is not an user alarm, but more a programmatically scheduled event) in intervals to perform its sync.

But some apps start a long-running service that can't be disabled by the user, and if you have many of these running, your performance will in-fact degrade.

1Google Music seems now also to run all the time. :( But there are other good examples of apps, that understand that they shouldn't run all the time. Or where the user is able to deactivate the background service. Or where the app developer was competent enough to make use of the rich (push) API of Android provides to keep the impact of the app at a minimum.