I've been running automated tools SEO tools for the past 4 years, pretty much 24.7.365 and you will be surprised how little you can get away with.
Everything that you said, you can easily get away with on any kind of VPS out there with 2GB of RAM and up with any processor post 2009.
The issue that you need to worry about running from home is not your servers resources, its your bandwidth. Depending on the sophistication of your upstream provider, they may develop a footprint with all that traffic and decide to throttle all of those outgoing packets to make room in the tubes.
The reason why most people opt to go for a VPS/Dedicated remote desktop is not a lack of local resources, it is because they treat it as a disposable resource. If you run into issues, get an IP blocked for abuse, or whatever it won't effect the way your room mate watches Apple TV, know what I'm sayin?
On the other hand if you do want to make the plunge, you can definitely get a box on ebay/or even new on Newegg for under $500 that is considered a "server." Just make sure that your processor has all sorts of multi-tasking abilities, multiple cores, hyperthreading etc.
The problem that a lot of people run into when working with automated tools is not the tools itself, its the lists.
For me, my lists get up to 500Mb in size easy - so working with/appending to different lists etc can freeze up your box way before running the tool itself will.