You wrote there is no way to change the hover color for post in the home page separately for different categories in the home page..but is there a way to change it in each category page?

Yes. View the page source for each type of page and look at the classes listed in the body tag. Each type of page has different tags, and you can use those to target the pages with different CSS.

Let's look at the projects page as an example:
http://silviacasaliphotography.wordpress.com/category/projects-progetti/

It has a body class of "category-projects-progetti"

So you can change the post hover color on those pages by using a category selector in CSS like this:

.category-projects-progetti .box .texts {
	border-color: red;
	background: red;
}

You would then add a new block like that for each different selector you wanted to change the post hover color for.

3. display a different featured image background color when image hovered for posts belonging to different categories (homepage)

There isn't a way to change the hover color for posts on the home page separately for different categories because of the way the HTML is structured. You could change them all at once with something like this though:

.box .texts {
	border-color: red;
	background: red;
}

Change the color names to color codes of your choice.
http://0to255.com/

2. create square thumbnails in the homepage as in http://robinlennoxstewart.com/

That is achieved by saving a square version of the image and setting it as a featured image for each post.

1. remove/hide "Category Archive/.." label and keep only the name of the category visible;

I see you found the solution for this one:

.category .page-title {
	visibility: hidden
}
.category .page-title span {
	visibility: visible;
	float:left
}