Some will run automatically, if they're either set to in their Settings, or just because they are installed.
I use a Task Manager, (something like Advanced Task Killer or TasKiller Free), to Kill All of them as soon as my phone starts up. They could be running, but the processes could be taking almost 0 resources.
Either way, those resources that the running apps take up are RAM. What you're running out of is 'ROM' (misnamed). It's where I described in Settings > SD Card and Phone storage.
You can have 2 apps. If the apps are 20MB each, or have HUGE caches, they can take up as much space as 20 apps or 200 apps. It's not about how many. It's about how much space. Some apps are just 50-100k. Some apps are 5-10MB. Pay attention, before you download them in Market.
I would make sure I have 40MB free space. I don't know what the low limit is there, but that's what I'd do.
My point kind of was, more importantly, that Swype probably isn't your problem. It DOES store a lot of data (I don't know if it calls it cache or data), and gets big (10mb++) because it learns your swyping. So it can act like you installed 10 more apps (in terms of how much space it takes up). Same with Browser, which is why I suggested to clear the cache from Browser.