I think I understand what you're asking. Let me see if I can rephrase this, to help others help you:
* You already are able to read music, and have been studying/playing classical piano for 8 years. (I only mention that because what you are asking is related to the frequent questions of "playing by ear" which usually meet with the response "go learn to read music" from many on this forum, including myself...)
* You want to be able to sit down with a fake book/lead sheet, where only the melody and chords are given and make each piece ("song" or otherwise) sound like real music rather than just a clump of notes on the piano and a sung melody.
* You seem to have two main concerns in this issue:
(1) "common order of playing the thirds, fifths, and bass of a chord," which is commonly referred to as chord voicing -- you know what notes should be played when the fake book says "Bbmaj7" (Bb,D,F,A), but you don't know what order to play them. Is Bb the lowest note? Should D be on top? Or A? F?
(2) What should you do rhythmically with your hands once you've figured out an appropriate voicing for each chord?
There are many, many ways you can approach question #1. I'm not an expert in jazz piano, by any means. I'd recommend you to two books:
Mark Levine's "The Jazz Piano Book"
Frank Mantooth's "Voicings for Jazz Keyboard"
But even without those books, it's important to know that there is rarely a single "correct" voicing for any chord -- you can make almost anything work. A good starting point might be to keep the root as the lowest note of the chord at all times (there are exceptions, but it's not worth getting into that just yet) and keep the melody note as the highest note in the chord. You don't necessarily have to play every single melody note -- maybe just play the melody note that happens when the chord changes. Let the rest of your fingers find the other notes of the chords (in between the root and melody) with what feels physically comfortable. When I do this kind of playing, I find it's more comfortable for me to play three notes in my left hand and two in my right hand, but that's just me. When you go to the next chord, rather than starting over, you should think about where your fingers already are -- and go to the nearest chord tones.
Example: Bbmaj7 - F7
Let's say you play your Bbmaj7 chord like this (from bottom to top): Bb-F-A-D. Then you can play your F7 chord like this: F-Eb-A-C.
D --> C
A --> A
F --> Eb
Bb -> F
Notice how, other than the lowest note, no chord member moves by anything bigger than a whole step. Nice!
There are many, many other ways to approach this. Google "piano chord voicings" and see what comes up.
As for question #2, this is much harder to describe in words. It's mostly a matter of knowing the style you're playing to know what rhythmic patterns will work the best. Part of the challenge is that, when you're playing from pop music books, you're often trying to cover the roles of drums, bass, guitar, and keyboard all on the piano -- it's impossible!
I'd start by keeping it really, really simple. Just play the chord on every beat while you sing the melody -- think of your piano as being a metronome. As you get comfortable with this approach, you can start highlighting important accents/syncopations in the melody. Or, maybe you want to have your left hand keep a steady metronome beat and let your right hand follow the rhythms of the melody. That's another option.
Anyway, I hope that my answer has helped you, but also I hope that it helps other people (who know way more about this than I do!) be able to understand your question a little more easily.
Good luck!