It comes with time and practice. You can study music for the rest of your life, and in the end will only know a fraction of what there is to know (IMHO). Enjoy the journey.
I generally conceptualize all four notes of 7th chords simultaneously. I learned them in college theory classes, where I was required to play them on piano in different inversions, and also do class composition projects with four-part voice leading. Thinking about them one note at a time is sort of a bass centric thing, since bass usually only plays one note. IMHO learning a bit of piano is one of the best things you can do if you want to be a serious musician. Piano will help with both your grasp of theory and also with developing your ear.
I do think of chord extensions and alterations sort of like add-ons. But if I practice a compositions with these elements, I start to think of them in a more homogenous way. Actually, at this point the chord tones are so internalized, that I don't always need to think of them. So I am more free to think directly about the extensions and alterations.
When you first start developing walking bass lines you tend to think in root position. But as you get more advanced you can create more interesting lines if you can freely work in 1st, 2nd, and 3rd inversion. It's a matter of knowing the notes, where they are located on the instruments, as well as the box shapes required to play them.
As you get more developed you also start using non-chord tones. So at some point you will also probably want to practice modes a bit. Modes are sort of like scale in inversions. In other words it's the same exact notes, you just start at a different point in the scale. If you are in Major, then Ionion is 1, Dorian is 2, Phrygian is 3, Lydian is 4, Mixolydian is 5, Aeolian is 6, and Locrian is 7. The same modes are used for natural minor but 6 (Aeolian) becomes 1. (Aeolian is 1, Locrian is 2, Ionian is 3, Dorian is 4, Phrygian is 5, Lydian is 6, Mixolydian is 7)
The Harmonic and Jazz Melodic scales generate a set of altered modes with modified or new names. You need to be pretty advanced to really keep them in your head; and FYI I really can't, although I do review them from time to time.
There are some songs that are modal, but most of time, most people do not really think in modes. However, I think learning and practicing the modes is still a good idea because it teaches you different finger patterns that you will use as an advanced player. It also exposes you to new and useful sounds. If you know your modes, it's easier to play diatonic non-chord tones.
You might want to save these links for later:
MAJOR SCALE AND ITS MODES (labdoo.org)
HARMONIC MINOR SCALE AND ITS MODES (labdoo.org)
MELODIC MINOR SCALE AND ITS MODES (labdoo.org)
...or find your own links that are better

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If I have not made it clear: I view modes mentally as diatonic notes in a key center, the hand shapes that are used to play the notes, and also each mode has a certain sound. So learning the modes thoroughly involves at least three mental processes.
The intent is to learn multiple ways of conceptualizing the same knowledge. Then as you play, you move in and out of different mental processes depending upon your fluency in the required knowledge and which mental process translates the information most efficiently. For example, when you don't know the note names of a chord in a difficult key, you draw on your knowledge of box shapes associated with the chord type. Don't get the idea you can learn only the boxes though. You will advance further if you know the actual chord tones and how they fit, or don't fit in the current key center.
At some point chord symbols become archetypes that you recognize instantly and then automatically play with very little conscious effort. You are not necessarily totally free at this point, because what comes out is basically all of the licks and patterns you have learned, practiced, and committed to memory. Your memory serves as sort of a library of these musical elements. You see a chord symbol and simultaneously imagine in your minds ear the sounds you want while playing the notes on the instrument. You do this while anticipating the next chords in the progression and what you want to do with the line. So not only are you using various mental processes in real time, you are also anticipating what is coming next. At this point your playing is essentially instinctual.
As a really advance player you can start a solo line on any note and make it sound great. Learning chord inversions and modes is sort of a steppingstone to get there.
However, whether you should learn this material depends on your interest level and goals. If you are satisfied hammering out roots, you probably won't want to bother.
Good Luck!