Note: I haven't played in a few months.
STO has an entire huge multi-season campaign that you can play through, and you definitely don't need to pay to do that. It takes place a few decades after Spock tried to prevent the Hobus supernova from destroying Romulus (and failed), so it ties into the Star Trek (2009) movie but is effectively the timeline of the non-altered universe (basing this on the possibility that the time travel in that case forked off an alternate reality instead of replacing the existing one, for whatever reason).
Cons: Ground combat is terrible. It comes down to "lock autotarget, spam abilities, pray the other group dies first". Think WoW, except much less ability depth, and you're trying to manage 5 characters at once. And trying to use "FPS mode" is nothing but a very elaborate way to commit suicide.
I agree with basically everything you've said except for this. I use FPS mode exclusively during ground combat and do well. Of course, I also have all my keys remapped so that the entire left side of the keyboard controls all my abilities (holding down a modifier key makes it act on the right half of the abilities instead of the left half). Then I use mouse buttons for movement instead of the keyboard (as I have a plethora of mouse buttons). Of course that's not strictly necessary, it was just faster than clicking the buttons or trying to use the number keys to activate a few of them.
(What's this about managing your other characters? That's possible? (Obviously in space you can map their abilities, but I couldn't on the ground))
How hard ground combat is depends strongly, from what I've heard, on whether you picked any ground combat abilities when you were making your character. If you didn't, it will be harder for you than it will be for someone who had.
You get free ships as you increase in rank. Not sure if you get one for the last ship category when you advance to the last rank category. I think you might not. (I have not reached it, as I haven't played recently. I've played through the romulan season and the 'seasons' before it, and that is as far as I have gotten in the storyline. You can also go back and replay missions, and will get the mission rewards again, at a higher level if the rewards' level depends on your level when you complete the mission. Some rewards are at a fixed level and a few rewards don't have a level or level-scale with you as you continue to level up.)
I have used the auction house, and it is frequently useful for selling off loot which you don't want (especially green (uncommon?) or higher rarity, which bind on equip), although some things will simply not sell (or are selling for ludicrously low prices, especially often common (white quality) items or lockboxes), and in that case you're better off selling it to a vendor or just recycling it using the replicator (which gives you less money, I believe).
You can trade dilithium in the dilithium exchange to get the points that are used to buy the premium ships, or buy anything else that costs points (additional inventory or bank space, character slots, etc), however, it takes a LOT of dilithium and there are other things that require dilithium (certain crafting components for higher-end items, for instance).
Also, don't buy keys for lockboxes. The odds of getting a ship out of one are close to zero.
(I always got my dilithium from Ker'rat but I'm reading that it has been nerfed or the mission removed?)