I have a small LAMP application that I want to move to Amazon
I will need Amazon EC2 for this, right?
Yes.
I want my static resources delivered from a CDN. Does that mean I need Amazon's CloudFront along with EC2? (Or is EC2 also a CDN somehow?)
CloudFront is a CDN, but you can use any other CDN just like you would with a site hosted on your own servers. EC2 is not a CDN.
I would also like to use a PHP accelerator (probably APC), memcache, etc. Will I be able to do this on Amazon?
Yes, you have full root access to the server. You can install any software you'd install on a server you owned yourself.
Will I still be able to use MySQL? (I ask this because it seems that Amazon also has their own RDBMS; not sure if I am forced to use theirs)
Yes. RDS is neat, but you can always run MySQL on your EC2 server.