How do I speed up Android on Sony Ericsson Xperia X10 Mini? My first Android device is an Xperia X10 Mini. I've installed a custom ROM (miniCm7) and now my device is definitely 'cool'.

The only problem is that a lot of applications run in the background, slowing down the phone. Xperia x10 mini has a 600 MHz processor and only 256 MB RAM, so it can manage few applications altogether, despite they are applications I need (for example: WhatsApp, Facebook, Dropbox, and more).

How do I speed up and maintain it clean?
Applications don't run in the background, they're suspended and don't slow down the phone.  Services that run constantly might slow down the phone, and I believe Facebook and Dropbox both run services.  (See this answer for a bit more info on how Android manages resources.)

Your best option is to uninstall the apps you believe are causing issues.  Another thing you can do is install a custom kernel that will allow you to overclock the device (at the expense of battery life and assuming CM's kernel isn't overclockable), but that may not help very much.  Most phones with specs in that range don't run things all too well, which is how the manufacturer get people to spend more money I guess :P

See also: What are some general tips on improving overall performance of an Android phone?