Learn Hardcore Russian: Давай посидим в тихом парке и поговорим о планах на завтра. - Let’s sit in the quiet park for a while and talk about plans for tomorrow.

Давай here is a very common conversational way to suggest doing something together: “Let’s …” / “How about we …”.
Grammatically it comes from the verb давать (to give), but in this pattern it functions like a particle introducing a proposal. It’s informal and friendly.

Why is it давай посидим and not давай сидим?

Because давай + verb typically takes a perfective verb when you’re proposing a concrete action as a whole event.

посидим is perfective (from посидеть) and means to sit (for a while) / to have a sit.
сидим would be the present tense of сидеть (we are sitting), so it sounds like a statement, not a suggestion.

What’s the difference between сидеть and посидеть (and посидим)?

сидеть (imperfective) = to be sitting / to sit (focus on the process or state).
посидеть (perfective) = to sit for a while (focus on a limited, complete activity).
So посидим means “we’ll sit for a bit” / “let’s sit for a while.”

Is посидим present tense or future tense?

It’s simple future in form, because it’s a perfective verb. Perfective verbs don’t have a true present tense; their “present-looking” forms are used for the future.
So посидим = we will sit (for a while), but in the давай + perfective pattern it functions as Let’s sit…

Why is it поговорим and not говорим?

For the same reason: давай + perfective is the natural way to propose an action as a single event.

поговорим (perfective, from поговорить) = we’ll talk (for a while) / let’s have a talk
говорим (imperfective) = we speak / we are speaking (more like describing what’s happening, not proposing it)

What case is в тихом парке, and why does тихий become тихом?

В + location (meaning in/at) uses the prepositional case.

парк → в парке
adjective тихий must agree with парк (masculine singular) in the prepositional case: в тихом парке.
Ending -ом is typical for masculine/neuter adjectives in the prepositional case.

Why is it о планах, and what case is that?

After о (about), Russian uses the prepositional case.
планы (plural) → prepositional plural планах: о планах = about (the) plans.

Why do we say на завтра and not о завтра or в завтра?

Russian usually expresses “for tomorrow” with на + accusative: на завтра.
Here it means the plans are intended for tomorrow (a target time).
Also, завтра as a word is special: it often stays the same form, but with на it forms the set phrase на завтра.

Is the word order fixed? Could we rearrange it?

The given order is very natural, but Russian word order is flexible. You can move parts to change emphasis:

Давай в тихом парке посидим и поговорим о планах на завтра. (puts the place earlier)
О планах на завтра давай поговорим в тихом парке. (emphasizes plans for tomorrow)
The grammar (cases/endings) keeps the meaning clear even when order changes.

How formal is this sentence, and what would be a more polite version?

This is informal because of давай (friendly suggestion).
A more polite/neutral option (especially to someone you don’t know well) is:

Давайте посидим в тихом парке и поговорим о планах на завтра.
Давайте is the you-plural/polite form and softens the suggestion.