Bulbapedia states that using the Running Shoes does increase the wild encounter rate. Furthermore, in the games (at least in Pokémon HeartGold I'm sure) it says something along the lines of running in the grass creates noise and startles more Pokemon.
Anecdotally, when I walk through a patch of grass, most of the time I do so without encountering a single Pokemon. But running through the same grass I almost always encounter at least one wild Pokemon.
In the more recent games, there are NPCs that you can talk to and they will tell you what type of Hidden Power your Pokémon has. According to Bulbapedia, the locations of the NPCs are:
- Platinum: Veilstone Game Corner
- Heart Gold/Soul Silver: Celadon Game Corner
- Black/White: Mistralton City
- XY: Anistar City
- South-Eastern-most house
- Also the location of the Hidden Power TM.
- Omega Ruby/Alpha Sapphire: Fortree City
- Top row of houses, second from the left.
- Also the location of the Hidden Power TM
If you want to find out through combat, here's something I threw together (Disclaimer: there's likely a much more efficient way to figure it out):
1. Battle a Patrat / Lillipup.
Super effective: Fighting. Didn't affect: Ghost. Normal effectiveness: goto 2
2. Battle a Cottonee / Petilil.
Super effective: goto 3. Not very effective: goto 4. Normal effectiveness: goto 5.
3. Battle a Klink.
Super effective: Fire. Not very effective: goto 6. Didn't affect: Poison.
4. Battle a Roggenrola.
Super effective: goto 7. Normal effectiveness: Electric.
5. Battle a Sewaddle.
Super effective: Rock. Normal effectiveness: goto 8.
6. Battle a Woobat.
Super effective: goto 9. Normal effectiveness: goto 10.
7. Battle a Ducklett.
Normal effectiveness: Grass. Not very effective: Water. Didn't affect: Ground.
8. Battle a Sandile.
Normal effectiveness: goto 11. Not very effective: Dark. Didn't affect: Psychic.
9. Battle a Ferroseed.
Normal effectiveness: Ice. Not very effective: goto 12.
10. Battle a Timburr.
Super effective: Flying. Normal effectiveness: goto 13. Not very effective: Bug.
11. Battle a Darumaka.
Normal effectiveness: goto 14. Not very effective: Steel.
12. Battle a Drilbur.
Normal effectiveness: Dark. Not very effective: Rock.
13. Battle a Trubbish.
Normal effectiveness: goto 15. Not very effective: Poison.
14. Battle a Deino.
Super effective: Dragon. Normal effectiveness: Normal. Not very effective: Dark. Didn't affect: Psychic.
15. Battle a Blitzle.
Normal effectiveness: Dragon. Not very effective: Steel.
My thought process (possible types at each stage) is in the revision history of this answer.
Best Answer
If I recall, the Level of your pokemon cannot exceed certain limits until you have earned certain badges. Most of the times your traded pokemon out-level all the others on your roster. It is sometimes useful to limit their growth so that they dont go above their obedience limit.