No. Extension tubes that offer AF capability simply mean they have contacts that allow for electronic communication between the body and the lens. They do not have autofocus motors in them or add autofocus capability to a body/lens combination that doesn't have it without the extension tubes.
I will just address your first question here - your second question is off-topic.
Your camera will already tell you if the image is in focus: a green circle/dot will appear in the bottom left corner of the viewfinder.
If you want more precision, you can turn on the rangefinder option in the settings menu that will turn the exposure meter (in P, S, and A ...
As the question you've linked to says, this is a limitation of the camera. A workaround would be to flip the lens into manual focus after you get what you want with the focus-and-recompose method. That's an annoying extra step, but at least it will work.
Yes the camera will AF if you have it in a mode that will AF when you press the button. It's a remote button. That's all.
I found with my remote it more annoying that if would AF so put it manual mode so it would not focus hunt. But I was not take bird photos but photos at night where there was not enough light to AF.
Sadly, you are (probably — see end note) out of luck, at least unless you do so with hacking or spend some money on different gear. That's because the Sigma flashes are TTL-only, and cannot be triggered by shorting the center pin in the standard way. I know, that sounds so stupid that it's hard to believe, but there it is.
So, what can you do? Well, first,...