What kind of software, operating system, database, processor, architecture, antenna used for communication back to Earth, basically what computer does it use.
You've asked 6 questions there, of which at least two seem unrelated to the title so I'm going to ignore them.
This document describes the "Integrated Science Instrument Module". The OS is listed as VxWorks, and the processor architecture is just described as "PowerPC". A separate presentation, "Lessons from 30 Years of Flight Software" mentions it explicitly as a RAD750.
This is not the only computer though. There's also the "Command And Telemetry Processor" or CTP, which handles spacecraft-specific operations like "pointing in the right direction". It also is the bit that deals with communication between Earth and the instrument package. Unfortunately, I seem to be quite unable to find anything out about the CTP, other than there was a faulty one that was replaced by Northrop Grumman before the telescope was launched. The odds that it also runs VxWorks on a radiation-hardened PowerPC chip are good (because nearly everyone does that sort of thing, as it happens) but I can't get any confirmation of that so it remains a guess.
The software running on it is (unsurprisingly) a bit old and weird. According to JWST: Maximizing Efficiency and Minimizing Ground Systems:
Being developed in the early noughties, there was lots of enthusiasm for XML as a storage and interchange medium. James Webb Space Telescope XML Database and CCSDS XTCE goes into more detail (and includes delightful quotes like "XML is the ASCII of the future"). CCSDS stands for "Consultative Committee for Space Data Systems", and XTCE "XML Telemetric and Command Exchange".