Learn Hardcore German: Wir begegnen einander oft im Park und grüßen freundlich. - We often run into each other in the park and greet politely.

begegnen means to encounter or run into someone, typically by chance. So Wir begegnen einander oft im Park suggests unplanned encounters.
By contrast, (sich) treffen implies a planned meeting: Wir treffen uns oft im Park = We meet up (by arrangement) often in the park.
English “meet” is ambiguous; German distinguishes the chance encounter (begegnen) from the arranged meeting (treffen).

Why is it einander here? Could I say uns instead?

Yes. einander is the reciprocal pronoun “each other.” uns can also express reciprocity, so Wir begegnen uns oft im Park is also correct.

einander makes the “each other” meaning explicit and is stylistically a bit more formal/clear.
Do not combine them: not ✗ Wir begegnen uns einander.

Which case does begegnen take?

begegnen governs the dative: jemandem begegnen.
Examples: Ich begegne ihm, Sie begegnet ihnen.
With einander, the form doesn’t change, but it’s still dative in this sentence: Wir begegnen einander … (dative).

Is und grüßen freundlich okay without an object? Should it be und grüßen uns freundlich?

In careful grammar, grüßen is transitive (you greet someone), so adding an object is very natural: … und (be)grüßen uns freundlich or … und grüßen einander freundlich.
That said, native usage often omits the obvious object: … und grüßen freundlich is widely understood as “we exchange greetings.” For learners, adding the object is a safe, clear choice.

Why is it freundlich and not freundliche?

Here freundlich is an adverb modifying the verb grüßen, and adverbs use the base form: freundlich.
The adjective would take endings only before nouns: ein freundlicher Gruß.
Compare: Wir grüßen freundlich vs. ein freundlicher Mann.

Why im Park and not in den Park?

Because it’s location (dative) rather than movement (accusative).

Location: im Park (= in dem Park)
Movement: in den Park (e.g., Wir gehen in den Park)

Is the adverb order oft im Park standard?

Yes. The common order is Time–Manner–Place, and frequency words like oft count as time. So oft im Park is natural. You can move elements for emphasis (e.g., Im Park begegnen wir einander oft), but Wir begegnen einander im Park oft sounds marked and is less common.

Does und cause inversion or change verb placement here?

No. und simply links two predicates sharing the same subject: Wir [begegnen …] und [grüßen …]. There’s no new finite-clause word order to trigger; you don’t repeat the subject unless you start a new clause (which would be fine too: … und wir grüßen freundlich).

Which auxiliary do I use in the perfect with begegnen?

Use sein: Wir sind einander oft begegnet.
By contrast, treffen uses haben: Wir haben uns oft getroffen.

Could I express the same idea with treffen or begrüßen?

Planned meeting: Wir treffen uns oft im Park und begrüßen uns freundlich.
Chance encounter using a more formal verb: Wir begegnen einander oft im Park und begrüßen uns freundlich.
Note: begrüßen is a bit more formal than grüßen but both mean “to greet (someone).”

Is einander declined for case?

No, einander is invariant; its form doesn’t change. The governing verb or preposition determines the case logically, but einander itself stays einander.
You’ll also see compounds like miteinander, füreinander, gegeneinander with prepositions built in.

Are there good alternatives to oft?

häufig (neutral, a touch more formal)
öfter (comparative of oft = “more often”)
des Öfteren (set phrase, somewhat formal)
öfters (colloquial; some style guides discourage it)

Does freundlich mean the same as höflich?

Not exactly.

freundlich = friendly, warm, kind in manner.
höflich = polite, courteous (focused on manners/etiquette).
In this context either could work, but freundlich emphasizes warmth; höflich emphasizes politeness.

Any pronunciation tips for begegnen and grüßen?

begegnen: three syllables be-GEG-nen; the middle e is long: be-geeg-nen.
grüßen: the ü is the fronted vowel (shape lips for “u” while saying “ee”), and ß marks a long vowel: grüü-sen.