Yes it is true that Windows 7 doesn't get on with some of my older software, the Serif software that I purchased back in the 1990's for a start. But, to answer your question, all listed below work with Windows 7 Home Premium (except for the server based software at the bottom of the page which doesn't need to!):
I have also purchased and installed the full version of tinySpell (version 1.9 at time of purchase) which is working hard to keep my spellings correct, also mentioned in that same article by Robin.
For image re-sizing / rotating / contrast & brightness alteration I use Easy Thumbnails from Fookes Software.
For the transparent backgounds and animated images I still use Gif Construction Set Professional 3 from Alchemy Mindworks.
For my drawings at the moment I'm using Windows Paint (version 6.1,included with Windows 7) which is more powerful than earlier versions. I may also use my new DTP software (Serif PagePlus X4) to draw images (there is an Export As Picture option).
A piece of software that I use "in the background" to check that links are not broken is Xenu's Link Sleuth which I have found to be indispensible as the site has grown. I forget who suggested it to me but it is certainly a time and trouble saver!
The menu is still handled by the Free DHTML Menu Builder from CoffeeCup.com. and their Free FTP program still uploads the site each week.
The Guest Book and Search Engine software are not programs that need to run on the PC. They reside and work on the server so they are the same as they used to be:
Guest Book: PHP Junkyard.com
Search Engine: Sphider PHP search engine
Since March 2011 the search engine has been changed as the host I moved to didn't support it. The new search engine was "supplied" by the host and is KSearch from . I must admit that it is a lot faster at indexing the site than Sphider was and is fairly easy to set up too.