Generally... when you have some 33% victory condition met, it is about time to quit anyways. At a time the game is little challenge and more chore.
When it comes to gas, I have done the following:
Notice when you select a planet, you got the quick buttons? One is "queue a nearby constructor to build a gas mine here" or something (if it is a gas source). Whenever I place a colony, I do that on a fuel source in the system. If there is none, I pick the closest I know. When I conquer a colony, I make sure to populate the gas giants in the system (I have of course killed the bases of the former owner). I very rarely take manual control of constructors, but I use this feature a lot. Pay a bit attention if the enemy kills lots of bases during war. Replace the fuel sources they killed...
As you get ready to switch to hydrogen, do the walk around your system and build hydrogen mines. Using the above method.
I don't find the few secs used to manual book mines for fuel sources tedious at all.
About only time I do manual constructor work is when I find an ultra rare source and need a mine. I grab whatever is nearest, send it, and queue it to build a couple of defensive bases afterwards. Then I just set it auto so it will do whatever it feels like when done.
The kaltor's don't resupply themselves as far as I know. Ones you got decent ships (and fighters), make sure to kill them as they show themselves. Maybe leave the huge crowds in gas clouds until that time. Pirates are annoying and hard to get rid of. As you get powerful, do try to cover as much area as possible with long range scanners, it should push pirates further away.
As for ships with fuel issues. If you follow the tip above it should not happen until you got very large fleets. Once you start using fleets with 4 capitals and 8 cruisers or such things, use resupply ships as much as possible for those. They can easily run gas mines and space ports dry if they arrive low on fuel.