One adjustment for you. Even if FTL travel is impossible, Relativistic travel is still possible. Accelerating to the point that you get time dilation for the passengers. While this won't help the population of the planets, it would allow people to potentially survive an interstellar trip. Think: Ender's Quartet, without the Ansibles. A good example of colonization like this would be The Songs of Distant Earth by Clarke

Without FTL travel, trade isn't going to be very useful. If you run out of water on Alpha Centauri, you're dead long before people on Earth can help you. On the whole, I feel like economics goes too fast for the distances involved, making interstellar trade not very lucrative for anyone.

I assume your no FTL travel means no ansible either?

@guildsbounty, very good point. Time dilation might make the generation ship model of trade work better in a universe where inhabited planets are widely spaced. However moving at relativistic speeds degrades the ability of the ship to gather materials - any contact between ship and asteroid is more likely to be in the form of an almighty explosion than mining. Which leads on to the general point that accelerating people to a speed where the benefits of time dilation can cut in is just generally difficult and dangerous. But please do expand on your comment if you feel inclined!

Generation ships as a way of life with every now and then a contact should be pretty much it. Likely the ship itself will whizz by with exchanges in knick-knacks being made in small vehicles.

Check out Neptune's Brood by Charles Stross. He has some ideas about trade with slower than light restrictions. Granted, his civilization is kind of post humanity, so if you wanted to travel to Alpha Centauri you would just send a copy of yourself by laser link and have it download into a pre-grown body at the far end.

would you believe me if I say that you can trade anything between Earth and any distance places at speed of light without conflicting any known law of physics, come on materials are everywhere an Earthling periodic table is no different from anywhere else therefore the solution lies in 3D printing technology! I'll leave the rest to your imagination lol.

What about en.wikipedia.org/wiki/Quantum_tunnelling ?

Read: Lockstep by Karl Schroeder. A society in which each planet hibernates for a large fraction of the time to give ships time to travel.

If one had an object with the mass of the Sun but a much higher density [high enough not to matter for purposes of the question], how much velocity could a space ship gather via sling-shot without having to get so close that differential gravity would rip it apart? Beyond the limits of navigational precision, what other factors would limit the use of multiple sling-shots for further acceleration?

@MichaelMcGriff what about it? That makes for fine diodes.

@supercat a dense body at some far distance will not help one bit, thanks to conservation laws. You need bodies for sligshot; e.g. spacecraft takes some orbital angular momentum of Jupiter's motion around the sun.

@JDługosz: A single dense object which is essentially motionless relative to the entity wanting the sling-slot wouldn't help, but what if e.g. fast-moving and dense objects started coming vaguely near our Solar System [e.g. 3x the distance of Pluto from the Sun] periodically, moving fast enough that the integrated effect of their presence wouldn't appreciably alter the orbits of the planets?

Check out Niven's "The Fourth Profession;" it handles this issue by saying that interstellar trade is a one-time event, essentially. Without spoiling it too badly, an alien race skips around the galaxy at relativistic speeds, stopping at inhabited worlds to trade, and then leaving with no intention of returning essentially ever. This frame of mind leads to some interesting revelations near the end of that story.

House of Suns by Alastair Reynolds has a setting like this. Basically, large ships holding trade goods travel between star systems at near-light speeds. Meet ups are planned hundreds of thousands of years in advance (and last decades) to allow ships to arrive from across the galaxy. The relativistic speeds allow for individuals on the trade ships to age more slowly than the rest of the galaxy.

I think you've worked this premise into a logically un-solvable situation. Pretty much every answer boils down to either trading information or nomadic trading. It's already been mentioned that the round-trip time for comms is so long, that information may well be value-less by the time it's traded (you need to know what the other side is offering, but when you do, you know what to look for and thus get the info yourself). Nomadic trading on the other hand is circumventing the lack of FTL - the goods travel slowly, but the trade happens locally, speed is irrelevant.

I almost posted this as an answer but it really doesn't qualify as one, thus I'm using comments. Essentially, by making FTL comms and travel impossible, you've made both comms and trade at interstellar distances so inefficient and ineffective yet so incredibly expensive that it is almost never going to be worth it. In order for a choice to be made, competing choices need to be inferior in some way or that choice needs to be special - the choice of interstellar trade in your case is always inferior to local information gathering and local production, thus it has no reason to exist.

Forgot to add that for nomadic trading, the distances between stops can still be forbidding - how do you know the goods you pick up in one star system aren't useless by the time you get to one where they have value? It could be decades and comms are too slow to tell you whether the goods will be needed without some near-perfect sociological understanding that provides you with accurate predictions of civilization states. And even then, war, destruction and acquisition through other trade or plain luck can thwart your incredibly costly mission.

This is not an answer, so I won't make it one, but I feel like going down this path is a classic mistake in space travel sci-fi -- trying to make it yet another iteration of terrestrial frontier exploration rather than what it is, which is something completely new and different. Impossibility of FTL actually makes possible all sorts of new things that aren't possible on earth, specifically, freedom from empire and central control. There's no way to exert control over something accelerating away from you at near the speed of light in unpredictable directions.

Maybe you're approaching the wrong side of the equation. We can't travel FTL, so instead we need to become more patient. A combination of elongated lifespans, time dilation due to travel, technological stagnation, and advanced predictive computer modeling could make the issue moot. Modern day humans might have to wait 1,000 generations for the colony ship to pass through, but if future humans lived 10,000x as long as we do, and had the ability to predict centuries in advance what goods they would need, the travel duration is no big deal. Basically: "a really long time" is relative.