Hey bennie!
I am sending you an email with some login data to try out lay theme and the addons for yourself.
With the lay theme carousel you can have slides with images and captions and a counter yes. By clicking on the image you can go to the next slide, yes. You cannot have different positions of the images depending on the slide no. For this maybe the fullscreen slider addon is better with the horizontal option activated. The fullscreenslider is different and i think it would make sense for you to try both addons. Also with the fullscreenslider on phone your content will be stacked, while with the carousel addon it will still be a carousel also on a phone.
Theres no support for an index page with all the images of a fullscreen slider or a carousel.
The site http://tomordoyno.com/ has a different url for each image. Its not really a slider, but just many subpages that contain an image. You could recreate this by creating a project for each image. (Maybe using a vertically centered image like here: http://laytheme.com/documentation.html#vertically-center-content)
Each project has a "project thumbnail". You could add the "project thumbnails" to an overview page by using the "+Project" button in the Gridder. This way you would have an index page with all images. For navigating between the images you could use the "Project Arrows" feature: http://laytheme.com/documentation.html#project-arrows
Contact details / Name can stay fixed, yes. The name can be the site title (Customize -> Site Title -> Site Title) and your contact details could be the site tagline (Customize -> Site Title -> Site Tagline).
Can I set/alter the positions of all the elements through the html/css editor?
No, this is better done using the gridder itself but any other css changes you can do.
Can I set/alter the positions of all the elements through the html/css editor?
No