1. I'm not sure what you mean by progressing after that point, at the moment you can't capture systems/stations/planets or whatever (although mods might allow you to do such things). Once you get to a late game or endgame point in terms of your fleet size, I find the fun is in trying out different fleet compositions. What I mean by that is experimenting with fleets that are say, all high tech or low tech, or mixing whichever of those you like the most, finding out which ships are best handled by the AI and exploiting that. As far as getting new ships and weapons goes, the quality of stuff you can get from markets is sorta like this: Open markets < Black market < Military market. To get nearly anything above frigate or fighter sized on the military market you need sufficient reputation AND a commission for that faction (which makes other factions that are not allied really dislike you).
2. Mods will mostly solve this problem, and maybe you are missing a lot of content that you don't see in the military markets? If you are indeed just bounty hunting, look out for the bounties that say "convicted of desertion" instead of "a notorious pirate", because they are fleets from factions other than pirates and will have a different variety of ships based on that, at least in my experience. Outside of military and black markets, these are the only way I can think of of getting ships that you might not otherwise be able to find, since you do sometimes have the opportunity to capture disabled enemy ships after combat.
3. One of the technology skills will give you access to an alternative speed boosting hull mod called Augmented Engines. I believe this gives you the same speed boost as Unstable Injector without making your engines take 4x as much damage. It also increases the travel speed of the ship it was added to by 1. The drawback is that it increases your fleets' sensor profile, but personally this never bothered me.
In response to the bonus question tips: Be wary of refitting your ships in open space, certain changes can tank the Combat Readiness of any ships you refitted, so best to refit only when interaction dialogs give you the option.