If theres one thing i've learnt over the short space of time i've been doing electronics, and i'm sure some of the golden oldies will tell you on this board, the best practise is to keep it simple and then expand upon that.
Yes it would be nice to have an ARM9 super processor churning out all kinds of things, but the question is, do you really need it? All PHP and ASP.NET do is expand upon the CGI technology (Common Gateway Interface). C# and Java are both interpreted languages, they heavily rely on a framework around them. Embedded systems are much more resilient to time processing, apart from the main stack operations, Microchip kindly offer a lot of assistance in the form of example source code when it comes to modern day applications for the PIC processors. I've even found some through hole versions of the dsPIC, excellent for prototyping.
The TCP/IP stack will fit and then some onto a PIC18F4620 processor, yes you need a level shift or go for an LF or a embedded chip as previously mentioned in this post.
A PIC24 would be a good start if you "have on lying around", but think about what you actually want to acheive as an end result.
It's a bit like going into a computer store and buying an all singing all dancing gaming PC just to run MS Word on or browse the internet for example, you could do the same on a lot less power.
My advice would be to start simple, then move on from there.
But you need to decide what platform you are going to build it on first, the PIC would be a good one, as it is a good way to show non-believers how powerful a PIC can really be (especially a PIC18).