pickup height
 

Does it matter how high the pickups are, as regards as closeness to the strings?

Yes, very much so. If too close, the sound will ge muffled as the magnets pull the oscillating string when you play. The way it was set when the bass was shipped was the optimum height. Now too high and not too low. Lower will be a less metallic sound, slightly.

Does it matter with the wood combinations? Because of my three basses, all three have different pickup heights, as well as string heights.

String heights are relative to pickup heights. You can be further away from the strings with the pickups without problems but too close, and you can get magnetic pull which will muffle the sound.

Neck/rod position and adjustments also play a role here.

First, I set the neck where I want it and then the bridge. I will go back to the neck adjustment if needed as well as the bridge and tweak the two till satisfied. Then, the last thing I do is set the pickups to match 'that' set-up. Different basses have different tensions for multiple reasons so even if the strings are at the same height, the pickups might be a hair closer or further. I put them down low enough 'not' to be in the way of the 'sonic oscillation' of the amplified string (I think I just invented a phrase!).

All my basses are made by you. I think I've been playing my 5TN for so long that I just became one with the setup and feel of that bass. The 5EG, still doesn't feel natural to me yet.

The EG is a lacquer finish and the wood vibrates differently than the oil finished TN. Also, the woods are different as well and that also makes them vibrate differently. Vibration is also part of 'feel' so two different basses with other common construction between them but they are not identical.

I think I have to bring my basses in.

Just call me when you are ready.