Whewww! That's a lot of information needed because there are no simple answers.
I would think a web designer needs to have skills in graphic arts. The main goal of graphics are to grab and please the eye. They provide the first "hook" to keep people on the website. However, too much and people leave without knowing what the site was about and move on to somewhere more conducive to finding the information they are looking for. To me, this is what makes a website successful: the mix of graphics and textual information.
If you're good and can land enough jobs, it can be very profitable. Some designers I know make a very good living with web design being their primary job. If you aren't committed to the field, however, you may find it difficult to get enough business to make it worth your while even as a supplemental income.
Software is optional. It can be very handy even if you can code a page from scratch. More and more sites are becoming a mix of dynamic scripting languages and static html. A good editor, such as Dreamweaver or Frontpage2003,would come in handy when dealing with those scripting languages as they have improved over the years for better support and debugging. A graphics editor of some kind would also be handy, even if you aren't planning on designing original graphics. Some manipulation is usually required to make an existing piece fit in its intended place on a web page. There are several software packages that would suit the needs of a designer. If [tab] were around he'd recommend the Gimp and I concur as it is a free program easily downloaded and installed. There are others, however, that can cost from a bit over $100USD to $1000+ USD. It will really depend on your needs in that respect.