Sending two men to negotiate with the ruling council is "conquering"?
The Tellerites were criminals, hence the "illegal" activity.
If the Organians were the country bumpkins they originally appeared to be, what would've been their choices exactly? Either way they were going to be forced to give up their way of life as a strategic piece of real estate in a Federation-Klingon war.
Kirk was using the Klingons as a chip to force the Organians to side with the Federation and be "host" to Federation forces.
All Tellarites everywhere are criminals?
Only the ones taking Coridan's crystals without paying any money to the Coridan government. Sarek's use of "Tellearite ships" is a standard monolithic/bigoted use of the word, as in all Tellarites are blustering fools that like to steal from someone. Not all Tellarites are thieves, but the Tellarite government may have been making enough on the crystal smuggling to look the other way. It's impossible to determine from the allegation as presented. My mistake is I should have said "The Tellarites in question were criminals, hence..."
What kind of occupation were you expecting the Orgainians to suffer from the Federation? Thousands, or even hundreds of red shirts wandering the alleyways forcing the Orgainians to give them ale? I disagree. I don't think the Organians were to be conquered. Offering a primitive people assistance would change their life, but their life is already changing due to the Klingon invasion.
However, if they (the Organians) were who they actually appeared to be it would have been a violation of the Prime Directive, as I don't think those sheep carts were warp powered.
Alternatively, if the Federation was only keeping Organia from the Klingons, the Enterprise could simply have bombarded the entire planet with photon torpedoes and possiblely fabricated cobalt bombs to rip away the atmosphere, thereby rendering it uninhabitale. And Kirk wouldn't have to negotiate that.