I'm curious: Which tab do you mean? The lower-left-corner of the screen, the one that shows a picture of the ship/world/base you have selected? How do you monitor construction progress from that panel?
Easy! Just select the item you want to monitor, and keep an eye on the box. You don't need to worry about getting lost in the shuffle if you use the forward/back buttons at the top of the tab. Very useful! It's like having bookmarks for stuff you wanted to keep an eye on. Also infinitely useful in battle, where hitting the back button is much easier than finding my fleet again.
And a further question. How do you defend all the mining bases and whatnot in your territory? Do you put a couple of warship on patrol around each base, or do you use fleets in defense mode, with overlapping circles? Or both?
It's very situational. In this case I lucked out and got close proximity to four different gas fields. The easiest way to make sure your little picket fleets don't become out-of-gas and useless is to put them right on top of a gas mine. Don't use a really big fleet for something far away, unless you really mean business and will keep an eye on that fleet. The first consideration when setting up your defensive network should be "What can I reasonably supply?" The Second should be "What needs to be defended?" Only then do I ask "How many ships?"
*As I said a few posts up, I also use the main construction screen to create a surplus of automated ships that are not part of any fleet. I make as many of these as I can reasonably afford to build, maintain, and fuel. That way when I need a fleet for part of my defensive network, I just pick from my military ships screen (filtered to fleets, looking for (none) and the ship type I want, and distance from the objective). There's no reason not to use posturing with every fleet, even the ones that are just patrolling a planet. At the very least, it will give you a visual marker on the overlay screen so you can check on things at a glance.