На следующей неделе мы собираемся встретиться в парке и обсудить план.

With time expressions meaning next week / during the week, Russian commonly uses на + prepositional (предложный) case: на неделе.
So неделя → на неделе, and the adjective agrees in case, gender, and number: следующая (fem.) → на следующей.

• на следующей неделе = next week (the coming week after this one). Most neutral/common.
• на будущей неделе is also possible, but often sounds a bit more “broad” or less anchored (depending on context).
• на следующую неделю is different: it usually means for next week (as a destination/target period), e.g., перенести на следующую неделю = postpone to next week. That’s accusative because it’s like moving something onto that time slot.

Собираться in the present tense + infinitive expresses intention/plan in the near future:
мы собираемся встретиться = we are going to meet / we plan to meet.
It’s not a dedicated future tense form; it’s a present-tense verb (собираемся) used with a future-oriented meaning.

The base verb is собирать (to gather/collect), but собираться is a reflexive form that often means to get together / to be about to / to plan depending on context.
Here it’s the “plan/intend” meaning: собираться + infinitive.

This is about aspect:

• встретиться (perfective) = meet (once), have a meeting (a single completed event).
• встречаться (imperfective) = meet regularly / be meeting (as a process).
  For a single planned meeting next week, встретиться is the natural choice.

Yes. встретиться is a reflexive verb meaning to meet each other / to meet up.
Non-reflexive встретить usually means to meet (someone) / to encounter as an object:

• Я встретил друга = I met (ran into) a friend.
• Мы встретились = We met up (with each other).

After в:

• в + prepositional answers where? (location): в парке = in the park.
• в + accusative answers where to? (movement): в парк = into the park.
  Here the meeting location is stated, so it’s в парке.

Because обсудить takes a direct object in the accusative case (what you discuss).
For an inanimate masculine noun like план, accusative singular looks the same as nominative: план.

• обсудить (perfective) = discuss and (typically) reach completion of the discussion.
• обсуждать (imperfective) = be discussing / discuss in general, without focusing on completion.
  In plans like this, Russian often prefers the perfective to present the discussion as a complete task: meet and (then) discuss the plan.

Both infinitives depend on собираемся:

• мы собираемся (что сделать?) встретиться и обсудить
  So it means: we intend to meet and to discuss. No extra verb is needed.

No comma is needed because it’s a simple sentence with a single predicate idea (собираемся) and two coordinated infinitives (встретиться and обсудить) joined by и. It’s not two separate clauses with their own finite verbs.

It’s optional. Russian often drops subject pronouns because the verb ending already shows the person/number:

• На следующей неделе собираемся встретиться в парке и обсудить план.
  Adding мы makes it a bit more explicit/emphatic (e.g., contrasting with someone else).