Asking questions is good, don't feel bad.
Taking it backwards, your system cannot run 10.8 Mountain Lion.
It can support Lion, however, the performance may or may not be great. Lion requires 2GB of memory (AKA RAM), and many have found it needs 4GB to run well. While you can install 4GB on your late 2006 MacBook, the system will only access 3GB. This article talks about upgrading from 10.5 Leopard to 10.7 Lion. Otherwise, you'll need 10.6 Snow Leopard to access the App store. If you want Lion, call the Apple online store, which in the US is 800-MY-APPLE and buy a download code, which you'll use on the App store. Lion 10.7 is $29 and Snow Leopard 10.6 is $19.99.
Your system can easily support OS X 10.6 Snow Leopard. Call the Apple online store to order it. 10.6 requires a minimum of 1GB, and 2GB, or more, will give better performance. OWC and Crucial are good sources for memory. This Apple note tells how to replace the memory, and this video shows you how to do that.
Now, question 1. Two keys to speed are RAM or memory and free disk space. Discussed RAM already. Disk space the general rule is that you need 10% or more of the space free for systems usage. When you've deleted files, you've probably emptied the trash. Some have found using OnyX has helped with performance (you want the Leopard version). With any disk maintenance program, you should have a good backup before using it.
See this FAQ on OS X performance.