They basically generate extra resources....a refinery pulls from all extractors at the planet it's built at as well as all planets within one jump...it's a fixed rate of income per extractor...

The whole 1/3 or 3/3 is related to quota...a single extractor can only contribute to 3 refineries, regardless of where those refineries are...

Generally speaking, refineries aren't worth building unless you are playing a mod that balances them...it usually takes a lot of extractors (on the order of 12 or so) to make them even competitive with trade port income...if you have a large trade chain, refineries will basically never outperform trade ports....then consider that even if you do have places good for refineries, there won't be very many of those places and thus the up front cost of researching the required tech makes them less economical than trade ports (which will be great everywhere)...

FYI, refinery income (like trade income) is not affected by allegiance...

Also, IIRC the TEC cargo technologies enhance both trade and refinery income, but the Vasari trade technologies only enhance trade...

You get an increase per trade port for each node added to the chain.

playerTradeNetworkDef
incomeRateBonusPerLongestNodeDistanceNode .075

This is a great thread on the topic. 

The formula is as follows:

N = number of trade ports

B = Base trade port income

C = Length of Trade Chain

O = Bonuses from other factors (research, etc.)

Trade income = N * (B * (1 + (0.075 * C) + O))

A real case scenario

N = 4 (four trade ports

B = 1 credit/second

C = 3

O = 0.15 (two trade income techs researched)

Trade income = 4 * (1 * (1 + (0.075 * 3) + 0.15))

Trade income = 4 * (1 * (1 + (0.225) + 0.15))

Trade income = 4 * (1 * (1 + 0.375))

Trade income = 4 * (1 * (1.375))

Trade income = 4 * 1.375 (1.375 is your credits per tradeport with modifiers)

Trade income = 5.5 credits/second

Chain is better but after a while you must break it in order to build more trade ports.. they work interstellar if you have a starbase on both suns with trade upgardes, and you have no gap (planet with no trade port of yours). It's not that complicated, longer chain, bigger income generated from each trade port placed. Good thing the income is increased when you put the trade port down, so it is not necessary to be built, the increase will be there.. like it is a functioning trade port..

Yes it does. 

To see the chain legs zoom out to see planets and hover mouse over credits. White line is your trade chain.... 

No, that is chain link 4. In my example you would have to have two trade ports on one planet, or one trade port not connected to the trade chain. If you make out a desert planet with trade ports you can in theory have 9 trade ports with a trade length of zero because they are all on one planet.

Well, its pretty hard not to get a trade chain at all. But I always try and calculate the best way to set up a trade chain given the planets I have and am likely to conquer.   As shown here, simply having 3 planets in a row with a trade port gives 22.5% extra trade income compared to having 3 trade ports on one planet. And you can set up a 3 planet trade chain even with a small empire. With really large chains, it gets even more ridiculous.

And while it has been a long time since I've done it, I do believe if you have a starbase with a trade port at your star which connects to your trade chain, and you build another starbase with a trade port at a different star, the trade lane will go interstellar. I could be wrong on this one though.

Yes. The trade chain bonus is applied to all trade income sources regardless of if it is in the chain or not. That's why it is such a powerful mechanic.