"How did you find out what and where was my problem?"
Since the problem affected only the main page and the June archive page, it had to be in one of your June posts. So I visited each one of these posts to see which one produced a layout error. Once I located the culprit post, I checked its sourcecode and searched for the specific HTML errors that can cause such issues. In this case you could also tell by looking at the post itself, because the frame of the messed-up captioned image looked different.
"Can you tell me why this happened? I did something wrong?"
You probably tampered with the captioned image in the Visual editor. After inserting a captioned image you must make sure the cursor is outside the caption area before doing anything else, and you cannot use any of the tools of the Visual editor to change something that includes the image or its caption.
"I'm [...] even more proud of my widgets especially of my awards."
Of course. When I said they are too many, I meant they contribute to the loading time issue, but notice I didn't tell you to do anything about them: what I recommended had to do with the posts only. (Actually the sidebar can also be improved, but not much.)
"I will have to pass (no offence)."
No offense, but you're wrong!
Images downsized by WP lose some color and sharpness - see here:
So you don't get best quality and detail when you upload a very large image, you get them when you upload an image that isn't wider than the displayed version. The images you upload are 3072px wide. The attachment pages you view when clicking on the images display the images at 848px. Each 3072px original is 3MB, while a 848px original would be around 330KB: that's one tenth the size. Do you realize what a difference this would make in terms of loading time and storage space? For largest possible on att. pages, best quality, and no waste of loading time and storage space, the ideal would be to downsize copies of your images to 848px wide, and upload/insert these copies instead of the huge originals.