It's not that they start better then you it's that they being AI's get to cheat a little bit, plus AI's multitask a hell of a lot better then you and I. I can almost guess that you're a turtle player in the sense that you don't expand quickly because you're busy covering your ass before you move on. Before you mess with anything, try a new strategy. It's called "get out there and grab some planets puto."
Basically, everyone is starting off with the same thing, send out your couple of scouts, build a colony cap ship as well as a colony frigate with a few escorts and send them off in different directions. Your cap can fight it's own battles by itself so let it go by itself. If you have options between a lightly defended planet and a heavily defended one, go after the light one first and don't waste the time on the heavy one until you have the ship count to make quick work of it.
You don't need to protect your home or any early grabs because no one has the ability to be producing an attack fleet much less anything that can lay down some orbital bombardment. Just grab and go, get yourself out of the hole population upgrade wise so you have a positive tax income, but don't do any high stage upgrades until you're maxing out in population or orbital structures. Your early expenditures will stay low and you can grab more planets and expand your fleet faster.
Later on if you're still feeling like you're getting the shaft follow these directions.
You can't do anything like how you're picturing it. Random spawns are based on what planets are listed in the galaxyscenariodef file, so in order to change the number of resource asteroids you get, you're going to have to hit up every single planet.entity file there is and change the numbers yourself. There are a lot of them so clear up a healthy chunk of time.
Changing these files is going to require you to have a way to change the files from binary to text and back. One option is Harpo's textbin mod tool, another is the less updated Dekhranic mod tool. Both do the job equally in this case.
Once you've got the planet entity files changed over to text, look for the section that follows the line planetResourceSetupInfo
This example is from the PlanetTerran.entity file
The important things to play with are the red green and blue numbers. I don't want to explain them because it's obvious what you need to do. If you can't figure it out, then... Nevermind.
When you're done, you can leave them as text or make them binary again and put them back in the gameinfo folder. Run your game and enjoy. You'll have conflicting info with online players though so whatever changes you make, it's advised to make them into a minimod if you play online so you can simply turn them off for that (in entrenchment and diplomacy only. vanilla is all or nothing.) If you don't know how to do that, then just keep a copy of the originals in a special folder, and keep copies of the modified files in another special folder and copy and paste as needed into the gameinfo folder.