As for types, I have found this for Diamond and Pearl, so I don't know how much this has changed. Here is a list of pokemon types with few weaknesses (it does rely on some pokemon having the right abilities).
No Weaknesses
1.Dark & Ghost - Still Vulnerable to the move Foresight and Scrappy ability.
2.Electric with Levitate
3.Dark & Poison with Levitate
4.Bug & Steel with Flash Fire
One Weakness
1.Electric (Ground)
2.Water & Dragon (Dragon)
3.Dark & Poison (Ground)
4.Normal (Fighting)
5.Normal & Ghost (Dark)
6.Bug & Steel (Fire)
7.Water & Ground (Grass)
As for choosing between similar type pokemon, players have different strategies when they do this. Some people choose pokemon with the best moves [which is a very relative term between players], others choose by type, or choose based on stat growth.
Then it depends what events you want to use your pokemon in. Because newer pokemon games have contests, special battle arenas, effort values to grow specific stats, and breeding, you may want to consider different types of strategies.
Best Answer
Yes, repeating damage from different sources stack. The example Pokemon will take damage from poison/burn, sandstorm, and Bind separately. This is easily reproducible.