1.  What are trade ports used for?

Trade ports well give your credit income a boost, the more trade ports the more trade ships you'll have and the more income you can get.

2. How many/at what planets should I build them?

You need atleast two trade ports in differnt planets to start a trade route.  Planets can have more than one trade ports and the effects are cummulative. Sometimes if you discover a planet bonus on trade, you may want to put lots of trade ports there.

3.  How long until broadcast centers start increasing my planets and nearby planets allegiance? 

They start as soon as they are built, but weather the culture spreads or not depends on the culture spread rates which is dependant on reasearch.

4.  How does the Monalith Cannon work?

Once built and charged up, you click on the fire icon, and zoom to the galactic view where you can select a "Known" planet as the target.  Then the cannon turns, charges up and fires... BOOM.

Hope that helps.

If you look through the SoaSE Wiki (I don't remember the URL off the top of my head), someone's written up an article detailing how planning a trade port chain can be more beneficial than simply spamming them on every single planet you own.