1. The higher your morale value the more you can tax people thus the better your economy. It also effects voting when you're a republic or having a voting society. If it's low in the red or like below 45 then you could get voted out of office and lose your politician bonus you got at the start of the game for some turns. Population is the key to success for invasions, if you have low populations the growth rate is slower, whereas huge populations can yield you a 1000 man troup transport and hardly hurt the planet population taxation and growth values.

2. Yes that is what range means how many LARGE squares away from your most forward or outward friendly planet/starbase you can travel. When you have like a .3 or .7 then that also is another portion of a part of the next big square, doesn't include all of it just part of it. You can tell your range by looking closely when moving and you will seen a pretty solid grey line outlining your maximum travel boundary.

3. Taxes and Soldiers is a big plus lol not to be so overlooked, and you determine the tax revenue with the tax slider, high morale (quality 70%+) equals (low taxation) speeds up population growth, likewise low morale (quality 50 or less) (high taxation) will slow it down. Entertainment improvements will also speed up population growth and improve morale at the same time thus allowing you to tax more when you need to. There's also a super improvement that will speed up population growth by 50% (think of it like a wonder from Civilization).

4. To move population around build troop transports or colony ships. This is what I do if/when I have problematic colonies who have outgrown their "entertainment improvements" or have maxed out their growth. You're going to need troup transports anyways so this is the best place and time to use them for removing population from maxed out or problematic planets. So yes you can move them from planet to planet or colony to colony rather as well as build up a nice invasion force of troup transports.