New to this game and enjoying it immensely. Been reading the forums and enjoying that too.
Welcome! It is indeed a great game.
What does a trade route do for relations?
Whenever a freighter of yours has established a traderoute to another AI, or vice versa, a positive relationsmodifier is activated. You can find these modifiers on the Foreign Stats/Policy screen, under the Report tab. Relation Factors include things like whether your diplomacy is higher (+/++) or lower (-/--) than that of the AI, whether your military is bigger (+/++) /smaller (-/--), and whether there is trade (+/++) or not (neither positive nor negative) between you and that AI. These modifiers change the way the AI feels about you, if it has more positive than negative modifiers towards you, relations may increase, if it has more negative modifiers than positive, relations may decline. Some AI have permanent negative modifiers such as "warmongers" which makes them harder to befriend and more inclined to be involved in wars.
For the trade modifier it doesn't seem to matter whether it was you, or the AI who established a trade route between you, nor does it seem to matter how many freighters there are going back and forth. Any traderoute between the two of you gives you a +, if you are the only civilization the AI trades with, this + changes into ++ (or maybe the ++ is reserved for the most important (= most money) trade partner, but I haven't focussed trade on any AI in ages, so I don't know). In short, establishing a trade route, makes you slightly more likeable to the involved AI, which improves relations or makes them decline less rapidly, making it easier/cheaper to avoid war.
My latest trick of choice is to find one trade route that looks most profitable and the maximum number of freighters along that one route. Is there any disadvantage/advantage to multiple freighters on the same trade route? If there is a relationship bonus with trading, is it multiplied? Theoretically, it maximizes the effect of any economic starbases along the way through multiple freighters passing the same starbase, but I agree with earlier comments in this thread that this is not a great income increase in the first place.
It seems you have broken your promise of just two questions! But I will answer anyway:
Your trick is the best may to maximize monetary tradeprofits, though I'd like to add that the earlier you establish a trade route, the more income it will generate. A second advantage is that it is generally easier to defend a single stream of traders, than having your freighters all over the place. It also makes it much cheaper to equip the traderoutes with starbases (though the extra income is pretty shabby, except perhaps on tiny maps). Whether establishing multiple traderoutes is more likely to earn you the ++ rather than the +, I don't know (see above). Trade won't grant you more than two +'s though.
As for the downsides of your strategy: It is very much putting all your eggs in one basket. If for some reason you end up in a war with that single AI, all your tradeincome is completely gone and your starbasemodules become useless. Also if the traderoutes come under attack and you are unable to defend, a single AI fleet can wipe out all your traders without having to cross the entire map to find them. Thirdly, if the destinationplanet or the planet of origin is taken, all the freighters vanish (even if the new owner is not hostile to you). Basically you'd have to establish that that traderoute is very secure, or it becomes a big liability.
The counter to these objections is obvious: if your traderoutes are spread out between many civilizations, the chances of you losing a couple due to unpreventable war are pretty high and defending all of your traders is also much harder when they are spread across such a wide range.
Basically you need to determine whether maxmized trade income is more important or less important than the improvement to diplomatic relations trade gives you. If you want to befriend as much of the galaxy as possible, spread out your traders across different civilizations and just send one to each capital (well defended, good population, and a civ that loses it capital will no longer require +trading in relations as their military is most likely very weak) until you've reached everyone. If you want to make as much money possible on your traders, send them as far away as possible, to a high population planet, and then defend this traderoute and the AI you're trading with.