images not clickable

Well to do this, you will first need to make a child theme and then make lots of edits to the theme files.
I would first make a copy of the parent theme, set up your child theme and then start altering your theme files.
You can just make copies of the parent files needed in your child and then edit them.

I can’t give you exact instructions, but you’ll have to just dabble with the files in your child theme and do a lot of coding.

Personally, I think that you are trying to do something a bit complicated which might frustrate your users and make them wonder what’s clickable and what’s not…

“if an image is displayed – no link” – “if text is displayed – link to the post”

That adds unnecessary strain on learnability and memorability
http://www.nngroup.com/articles/usability-101-introduction-to-usability/

So there is still something that I would like to do : erase the website title and menu on the home page only (it should stay on posts and pages).

Sure, but since you’ll be setting up a child theme for the below change, wait until you have the child theme.

Then, within your Child Theme style.css file, enter this;

.home #masthead {
 display: none;
}

I will put the links I want in the circles directly on the home page.

Try this http://ww.wp.xz.cn/support/topic/trouble-displaying-title-on-hover-themespun?replies=12#post-3981752