Adam,

 
>>1. The spell checker doesn't seem to work (I get a popup with an asp.net error message), I get the same issue if I try to edit templates from the templates dialog box.

Can you post your error message? Is your site online so you can post a URL?

>>2. What is the difference between 'Css Class' and 'Inline Styles' in the drop downs

<span class="redtext">Hello World</span> // this is using the 'Css Class' 

<span style="color:red">Hello World</span> // this is using the 'Inline Styles'

>>3. I want to be able to control the Css Classes which appear in the drop down and am looking for the simplest possible solution. I notice that by default it includes all dotnetnuke styles. If I want to limit this list to styles contained in a custom.css file how would I do this (i.e. disable default behaviour of populating dnn css classes)?

Yes, you can do that with setting Editor.AutoParseClasses property to false.

Editor.AutoParseClasses Property
Specify whether or not the Cute Editor should automagically parse the CSS classes from EditorWysiwygModeCss and populate all items into CssClass dropdown.

In the CuteEditor section of your web.config file, just add AutoParseClasses = "false"

Then Edit Dropdown Configuration file (Common.config).

The dropdown configuration file (Common.config) can be found in the /CuteEditor/Configuration/Shared folder. In dropdown configuration file you can find the CssClass element which contains the configuration information for the CSS Class dropdown within Cute Editor.  

For details, please check this article:

http://cutesoft.net/developer+guide/Class_dropdown_Customization.htm

Keep me posted