Firstly: Your old PC. Apart from an Antivirus program and Firewall in full operation, what housekeeping do you run on a regular basis? What you have described suggests a build up of detritus on your Hard Drive, which is relatively easily cured and can give an old faithful a new lease of life.
Secondly: You don't need a Custom Built PC for the work you plan to give it. There are plenty of mass produced High-end PCs on the market. Dell are a well thought of product, but there are a dozen or more companies trading in the UK that will have such a PC in their catalogue.