I have a $250 Toshiba i use for chess ,and it works fine.
They use plasic for the power adapter on thoes sell it as soon as you notice its loose, unless you are too nice..
Whether your computer has a cheap build that's another matter entirely. Strictly from a processing point of view, even a $250-Toshiba will be more than enough to aid humans with chess analysis.
Anything modern will be fine. No need to break the bank. I doubt they even sell laptops that don't have SSD drives these days. Those help with the overall speed of the computer and using endgame tablebases for engine analysis.
Indeed. Modern engines make use of several optimizations and heuristics strategies to cut on processing power. For example, I have a Samsung Galaxy Y 2 Pro with a 1 Ghz processor (1 core) and 512 MB RAM. That's a very crappy cellphone. Guess the ELO of Shredder here? 2500 ELO. This is comparable to a human grandmaster in performance. So, I suppose even a Pentium III, released in 1999, should give you good engine performance.
Since in practice you are not going to get a computer this old and probably want a modern operating system, I would suggest at least 2-core computer, so you can optimize your engine to run with one core and the operating system with another – that is, you will be able to distribute processing power evenly.