Learn Hardcore Russian: Раз в неделю мы встречаемся в парке.

Yes. The suffix -ся/-сь makes the verb reflexive. In встречаемся (from встречаться) it’s reciprocal: we meet one another. Without it, встречаем (from встречать) means “we meet/greet/pick up” someone or “we encounter” something, not “meet each other.”

Why is it the imperfective present встречаемся instead of the perfective future встретимся?

Russian uses the present of the imperfective to express habitual/repeated actions. Мы встречаемся = we meet (regularly). Встретимся (perfective future) refers to a single meeting in the future: Мы встретимся в парке в пятницу = we’ll meet (once) in the park on Friday.

Why is it в неделю and not в неделе? What case is that?

It’s Accusative after в in the “per unit” meaning. Раз в неделю literally means “once per week.” With frequencies you use в + Accusative:

два раза в неделю, три раза в месяц, пять раз в год.
Prepositional в неделе exists in other meanings (e.g., В неделе семь дней = there are seven days in a week), but not for frequency.

Why в парке and not в парк or на парке?

в парке (Prepositional) = in the park (location, where).
в парк (Accusative) = to the park (direction, where to).
на парке is incorrect; for parks Russian uses в.

Can I move раз в неделю to a different place in the sentence?

Yes. Word order is flexible:

Мы встречаемся в парке раз в неделю (neutral).
Мы раз в неделю встречаемся в парке (frequency in focus).
В парке мы встречаемся раз в неделю (place in focus).
Раз в неделю мы встречаемся в парке (time/frequency first; very common).

Is один раз в неделю different from раз в неделю?

They’re effectively the same. Одни́н раз в неделю adds explicitness/emphasis (“one single time per week”). Раз в неделю is the most common, natural way to say “once a week.”

How do I say “twice a week,” “three times a month,” etc.?

Use the pattern число + раз(а) + в + период:

2–4: раза → два раза в неделю, три раза в месяц, четыре раза в год.
5+ (and 0): раз → пять раз в неделю, десять раз в год. Also common: несколько раз в неделю (several times a week).

Can I drop the subject мы?

Yes. Russian often omits subject pronouns because the verb ending shows the person/number. Раз в неделю встречаемся в парке is fine if context makes “we” clear. Including мы is also perfectly normal.

Could мы встречаемся mean “we are dating”?

Yes. Встречаться can mean “to date.” Without extra context, Мы встречаемся is often understood as “We’re dating.” When you add time/place/frequency (в парке, раз в неделю), it usually means “meet up,” not necessarily romantic.

How is the sentence stressed and any pronunciation tips?

Stresses: Раз в неде́лю мы встреча́емся в па́рке.
Tips:

встр- is pronounced like fstr- (the в devoices).
ч is like English “ch,” but a bit softer.
-ся sounds like “-sya.”
Vowel reduction: the first е in встре- is reduced; keep the stress on -ча́- in встреча́емся and on -де́- in неде́лю.

Why does неделю end in -ю?

Неде́ля is a feminine -я noun. In the Accusative singular (used after в here for the “per unit” meaning), it changes to -ю: неделю. This is the regular feminine pattern: книга → книгу, неделя → неделю.

Can I say каждую неделю or еженедельно instead of раз в неделю?

Каждую неделю = every week (doesn’t specify how many times within the week).
Еженедельно = weekly (formal/literary; likewise doesn’t specify count).
Раз в неделю = exactly once per week.
Choose based on whether you want to stress exact frequency or just regularity.

Do I need to add с (with) after встречаемся?

Use встречаться с + Instrumental when you name whom you meet: Мы встречаемся с друзьями в парке. If the subject is plural and you mean “meet each other,” с is optional: Мы встречаемся в парке already implies reciprocity.

Do I need a comma after Раз в неделю?

No. Short introductory adverbials like Раз в неделю don’t take a comma: Раз в неделю мы встречаемся в парке.