Learn Hardcore Russian: На заправке я решила заправиться до полного бака бензина, чтобы потом не волноваться. - At the gas station I decided to fill up the tank with gasoline so I wouldn’t worry later.

На заправке is the usual way to say at the gas station in Russian. Many “places/locations” are idiomatically used with на (on/at), e.g. на вокзале, на почте, на работе, на заправке.
Here заправке is Prepositional case (used after на when it means location: где?).

Why is решила in this form? Does it show gender?

Yes. решила is past tense feminine singular of решить because the subject is я spoken by a woman (or a feminine narrator).
If a man says it: я решил.
Plural: мы решили.

What’s the difference between решить and решать here?

решить (perfective) = to decide as a completed decision (one-time result). That fits here: she made a decision at that moment.
решать (imperfective) would be used for an ongoing process or repeated decisions, e.g. я долго решала, что делать (I was deciding for a long time).

Why is заправиться reflexive (-ся)? What does заправиться mean compared to заправить?

заправиться is the common everyday verb meaning to get fuel / to fill up (one’s car)—literally “to fuel oneself,” i.e. your vehicle.
заправить is transitive: заправить машину (to fuel the car), заправить рубашку (to tuck in a shirt), etc.
So заправиться focuses on the action you do for yourself: “I filled up.”

Why is заправиться perfective? Could it be заправляться?

заправиться is perfective because it’s a single completed action (the goal is to finish fueling).
заправляться (imperfective) would be used for:

a habitual action: я часто заправляюсь тут (I often fill up here)
an action in progress: я заправлялась, когда он позвонил (I was filling up when he called)

What does до полного бака mean grammatically? Why is полного in Genitive?

до + Genitive means up to / until (reaching a limit).
So до полного бака = until a full tank (i.e. “to the point where the tank is full”).
That’s why полного and бака are Genitive.

Why is it полного бака бензина and not something like полный бак бензина?

Because до requires Genitive: до чего? → до полного бака.
Inside that phrase, бензина is also Genitive because it’s a “quantity/container + substance” pattern:

бак чего? → бак бензина (a tank of gasoline)
So the whole chunk is: до (полного) бака (бензина), all Genitive as needed.

Is полного бака бензина the only natural way? What about до полного бака without бензина?

Both are natural.

до полного бака is very common and usually understood from context (at a gas station it’s obviously fuel).
до полного бака бензина is just more explicit; it can sound slightly more “complete/technical,” but still normal.

What does чтобы do here, and why is the verb не волноваться?

чтобы introduces a purpose clause: in order to / so that.
After чтобы, Russian often uses an infinitive when the subject is the same:
я решила …, чтобы потом не волноваться = “I decided … so I wouldn’t worry later.”
не волноваться (imperfective) describes the general state/process of worrying, not a single “completed worry.”

Why is не separated: не волноваться?

With verbs, не is written separately: не волноваться, не делать, не знать.
(There are rare exceptions, but for normal verb negation it’s separate.)

What does потом mean here, and where can it go in the sentence?

потом = later / afterwards.
Its position is flexible. These are all possible with slight emphasis differences:

…, чтобы потом не волноваться (later = after fueling)
…, чтобы не волноваться потом (slightly more emphasis on “not worry”)

Is the word order fixed? Could I say Я решила на заправке…?

Word order is fairly flexible because endings show roles.
Both are correct:

На заправке я решила… (sets the scene first: “At the station…”)
Я решила на заправке… (starts with “I decided…”; location is secondary)
The original is a very natural “context first” order.

How do I pronounce заправиться and where is the stress?

Stress: заправи́ться (za-pra-VEE-t’sa).
Also: запра́вка (za-PRAV-ka), запра́вке (za-PRAV-ke).