First, it seems that the 'page title' is determined by the joomla sef urls...correct?
Kinda, that defines the URL, and if you don't set the page title then it will use it, but you can set your custom page title in the same place, it's located in the Parameters, System tab. The choose to show or not show refers to on the actual page itself.
I wouldn't worry about using a tool for your H tags, you can do that yourself and won't bloat your site.
The font size of the headings are quite large compared to the body text. is it okay to make them a smaller font and change the color or does that mess up the seo characteristic of the tag?
Sizes are and colors for the tags are defined in the CSS of your template, have a poke around and change them as you see fit. Use firebug to see where they are getting pulled from.
All of these things are what's called Basic On Page Optimization (or Optimisation depending on your country) They are just the beginning. But you must get the basics down before going any further.
A great resource to start learning SEO is a Free PDF called
SEO Faststart.
It's a great read. I am in no way affiliated other than I use it as a resource.
Hope that helps.
Cheers,
Rich