Here is the whole procedure you need to do, right from the beginning. I'm writing all this just to avoid any confusion between your initial intersects, and fine tuning your approach.
First thing here is to make sure your inclinations are as close as possible to eachother.
If you are leading the ship/object you want to dock to, burn prograde to increase your AP/PE. Your goal here is to make your orbit larger than the other ship's orbit. However, ensure that at some point, the 2 orbits intersect each other, and not at sharp angles. Try to make them as close to parallel as possible where they meet.
If you are trailing the ship, burn retrograde in order to make your orbit smaller. Again ensuring the orbits intersect at some point.
Don't go crazy with the difference in the two orbits, time and experimentation will give you more experience on getting this right. You can also re-adjust your orbits now and then by burning either prograde or retrograde at the point where the 2 orbits intersect, and seeing how it affects your intercepts.
To give you an idea how this works; a ship with a smaller orbit will make one orbit faster than a ship with a larger orbit. So a smaller orbit is "faster". So yes you were correct in your statement.
Then it's just a matter of time warping until you intersect. Make sure to right-click on the other ship/object in the map view and set it as the target. This will then show you orange and red arrows that indicate where, when, and how close you will intercept your target.
When you see the intercept arrows, you can stop the time warp and try adding a maneuver node at different spots in your orbit to try different things that may make your intercept closer. Check your distance by hovering your mouse over the orange or red arrow, make a small adjustment to the node and check to see the difference. Again, more experimenting.
Once you are within 0-10 km's (under 10 is my personal preference but it depends on your ship's abilities), you then want to click on the speedometer above the navball until it says "target" and a speed under it. This is your speed relative to your target, so if it shows 0 (zero), then you are moving at the exact same speed as your target.
Now you will point directly towards your target (purple on navball) and do a burn to get closer and raise that target speed. At this point you will see your distance from the target dropping. Now, you're moving towards your target, but your orbits are different so you will likely not hit it directly unless you were really close to begin with.
Turn your ship now to point at the yellow retrograde. Since you are in "target mode", the retrograde is adjusted in reference to your target and its orbit.
As you start getting close, you may find that you are passing beside it. This is normal. Still pointing at the yellow retrograde, you want to now burn and get your speed back down to zero. Then, point back to your target (purple) and repeat this until you are within 1KM (1000 meters) or less. Be aware that you don't want to get too close to your target when burning with a rocket because rockets can push your target away from you or send it into a spin.
Once you're within 1KM and you've brought your speed down to zero, you can now switch to using RCS only for the rest of the approach. RCS thrusters will not affect the other ship like rockets do so you don't have to worry about using them when you're close to it.
The most important part of all of this is to have patience and don't rush. If you pick up to much speed when trying to get closer, you could screw it up by overshooting your target. And it's normal to have to do several slower approaches to get closer and closer. The more you do this, the better you'll get at it.
Good luck!
Kerbals can't die for any other reason than lithobraking (just like aerobraking, only difference is that you replace the atmosphere with the ground) and a collision with an object.
When you're in control of the command module, try switching to Neil by using the [ and ] keys. Unless there's a lot of debris near by, it should only switch between the module, flag and any kerbals on EVA.
If taking control of Neil doesn't work while you are in control of another object, you can go back to the space center and enter the tracking station. There you will be able to select your kerbal from the list on the left and press the "Fly" button.
The only thing that could prevent you from taking control of that poor kerbal is a bug.
In this case, basic troubleshooting. Reload your save file. If that fails, restart the game. If this fails, well, assume that Neil met the kraken.
You could also try bumping him so some of its variables in the code are changed.
If all fail, well, roleplay it. He died of hunger while you pressed the big red button in the tracking station.
Best Answer
What you're looking for is called translation, and it is possible in KSP. But you'll need Reaction Control System enabled on your vessel to be able to do it.
You can use H, N, I, J, K, L (default keybindings) to perform short RCS thrusts in every direction. Or you could switch to docking mode (it's the small blue button in the bottom-left corner of the screen) and press the space bar to switch A, D, W, S between rotating the craft and translating it.