How to remove the clock icon?

Check your source! Your first selector is wrong. It should be:

.site-content .entry-meta .date a:before

Depending on where that code appears, you won’t need the .site-content.

Great, thanks…

So, for clarity, what worked in the end was that I added the following to the style.css of the child theme:

/* Hide clock icon from top of blogposts */

.entry-meta .date a:before,
.attachment .site-content span.entry-date:before {
display: none;
}

/* Hide Date from top of blogposts */

.entry-date {
display: none;
}

Related follow-up:
Because the clock and date fields are hidden, but not removed, their ′pixel space′ is still used, just appears empty, making the Category and Authors fields appear to be tabbed in a bit.

Looks like this: https://techblog.king.com/testblog/

The order of things is, from left to right, Clock, Date, Category, Author.

Would it be possible to change that to Category, Author, Clock, Date?

This would effectively left-align Category, Author.

Quick point of clarification: All these styles you’re wanting to change are from Twenty Thirteen and aren’t modified at all by Blogsonry. My goal for this theme is to change as little as possible from the parent theme.

That “tab” is coming from a 20px right margin on span.date. Since you’re only display: none-ing .entry-date, that span is still there with it’s margin.