How/where can I catch 'em all?

I'm missing about half of the pokémon, and I keep catching the same pokémon all over again — Abra, Meowth, Psyduck...

I noticed that if I catch over three pokémon I'll be able to go into the Meowth bonus stage, and if I get higher scores and more pokémon to other bonus levels too in some sort of progression — Meowth, Gengar, Mewtwo, etc. Does a similar process take place with catching pokémon, in which the further I progress without losing, the greater the variety of pokémon I'm able to catch? Or is there some other trick to it?

As for their distribution: they don't appear to me in a random order, I think. Are the pokémon distributed evenly throughout the various locations? If so, where can I catch each of them?

Catching

You're already familiar with this method but I should also point out that you can enter catch mode with either 2 or 3 GET lights lit up (whereas EVO mode requires all 3). Different pokemon will appear depending on whether you have 2 or 3 lit.

There are two main methods to obtaining more pokemon outside of just catching them: moving and evolving.

Moving

On the Red stage, hit the diglett 3 times. On the Blue stage, hit the psyduck/poliwag 3 times. This will activate Move mode.

When you're in Move mode, you will need to hit several targets (indicated by arrows) and then shoot your ball into the hole in the middle of the stage. Moving will take you to another location, where different pokemon can be caught. The red and blue tables have different map locations, with most of them being exclusive to one table or the other.

Map locations are split into several "pools" with rarer pokemon being found in later pools. The first three locations are chosen from pool 1, the next two are from pool 2, and the last being Indigo Plateau. After this it will cycle back to pool 1.

Evolving

Activating EVO mode is done similar to activating catch mode except you move the ball the opposite way around the big loop three times. You must have at least one pokemon caught in order to activate EVO mode.

Similar to move mode, arrows will appear on the stage in several places, and you need to hit them with your ball. Other items will then appear (such as evolution stones, EX for experience points, or a link cable -- whichever is appropriate), which you need to collect three of. Afterwards, shoot your ball into the hole in the center of the stage. Your pokemon will then evolve.

Source: bulbapedia and personal experience