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!
Is this just a limitation of the physics engine in KSP - that certain
factors such as aerobraking aren't taken into account for objects that
aren't explicitly focused upon?
Hit the nail on the head right there, objects that aren't the active vessel or within close range of the vessel are considered to be "on rails" by the game engine and there orbits will not be changed. This is to save on processing time, an assumption in the game mechanics is that, when the vessel is not within the active zone (500m around the active ship AFAIK), it is incredibly unlikely that they will interact with anything that will change their trajectory or velocity in anyway, so recalculating the orbit is unnecessary.
Easiest way to fix this is just to make the capsule the active vessel for awhile and use time acceleration to speed this up.
You can use physical time acceleration inside atmosphere by pressing Alt + .
Best Answer
With the new aerodynamics system it is quite hard to tell, because the ideal path depends on the aerodynamic drag, lifting surface to mass ratio and thrust to mass ratio. So you should experiment with your craft to find the ideal ascend profile.
But in general you should try to get to an altitude above 10km as fast as you can, which means a climb angle of about 40°. Oh, and don't forget to retract your landing gear.
When you reached the middle atmosphere (between 10km-20km) you want to gain as much horizontal speed as you can from your air-breathing engines as long as they can still get intake air, which usually means a very flat angle between 10° and 20°. The limit how flat you can go depends on the maximum speed the craft can still withstand aerodynamically without losing any parts. Monitor the thrust of your jet engines (right-click) and switch to rocket engines (closed-cycle mode with RAPIERs) as soon as their thrust becomes negligible.
During the coast-phase to the apoapsis, make sure your craft stays pointed prograde. There isn't much drag anymore above 40km, but when your craft rotates so that the wings are perpendicular to the heading, you will still lose considerable speed even in the upper atmosphere.