When independent clauses are not truly independent

Perhaps, you should have chosen a better example. Anyways.

Surprisingly, my car can go pretty far and it gets good mileage.

That is semantically incorrect (fails to make proper sense.) Here's why:

Surprisingly, my car can go pretty far and still (give) good mileage.
Surprisingly, my car can go pretty far and also (give) good mileage.

It is the sentence as a whole, not either of the clauses that takes the attributive surprisingly, so there's a coherence required between the clauses. Independent clauses stand apart, instead.

Check this: The verb agrees in number with can, not the nearer verb.

In speech and in informal writing, one could get away with anything, though.

HTH.

ExSum: Turns out I was merely restating Prof. Lawler's comment in essence.

Your problem:

You want to show surprise over two events:

"Surprisingly, my car goes pretty far."

"Surprisingly, my car gets good mileage."

But when you form a compound sentence with one "surprisingly", it only appears to modify whichever thought you place first.

This is because "surprisingly" is no longer attached to both thoughts.

---

Resolutions:

(1) The fact that the car "goes far" is not actually a surprising event, but just a secondary idea that you are attempting to fill in with the ACTUAL surprising event: "Surprisingly, my car gets good mileage [when I drive long distances]." I have now subordinated the secondary thought as a dependent (adverb) clause.

(2) You could show surprise in both independent clauses: "Surprisingly, my car can go pretty far, and [(also surprisingly,) (just as surprisingly,) (even more surprisingly)(etc...)] it gets good mileage." Now both of your independent clauses show your surprise.

(3) The easiest solution, keeping it as close to the original as possible, to show surprise about both events is to simply state both actions that your car performs to surprise you in one independent clause modified by the adverb 'surprisingly': "Surprisingly, my car can go pretty far and gets good mileage." Now one "surprisingly" modifies both verbs.