Learn Hardcore Dutch: Ik leg mijn notitie in de map en bewaar alles daarin. - I put my note in the folder and keep everything in it.

leggen – to lay something down in a horizontal position
- Ik leg het boek op tafel. – I lay/put the book on the table.
zetten – to put something so it stands (vertical / upright or on its base)
- Ik zet de fles op tafel. – I put the bottle on the table.
doen – to put in, but very general and often less precise
- Ik doe de brief in de envelop. – I put the letter in the envelope.

A notitie (note) is usually flat, like a piece of paper, so leggen is the natural choice:
Ik leg mijn notitie in de map.

You could sometimes hear Ik doe mijn notitie in de map, but leggen sounds more precise, especially for paper or documents.

What exactly does bewaar mean here? Is it like keep, save, or store?

Bewaren basically means to keep something so it is safe and not lost or damaged. In this sentence:

ik bewaar alles daarin ≈ I keep/store everything in there.

Common uses of bewaren:

Koekjes in een trommel bewaren. – Keep cookies in a tin.
Een e‑mail bewaren. – Save/keep an email (don’t delete it).
Een geheim bewaren. – Keep a secret.

It is not usually to hold in your hand (that’s vasthouden) and not to rescue (that’s redden). It’s mainly about keeping something safe or in a fixed place.

Why is there no ik before bewaar? Why not Ik leg… en ik bewaar…?

In Dutch, if you have the same subject for two verbs in a row, you can leave out the second subject:

Ik kook en eet. – I cook and eat.
Wij lezen en schrijven. – We read and write.

So:

Ik leg mijn notitie in de map en bewaar alles daarin.

is a shorter version of:

Ik leg mijn notitie in de map en ik bewaar alles daarin.

Both are correct. Adding ik before bewaar is a bit more explicit or emphatic, but not necessary.

Could I also say Ik leg mijn notitie in de map en daarin bewaar ik alles? Is that correct?

Yes, that is correct, just with a different word order and emphasis:

Ik leg mijn notitie in de map en bewaar alles daarin.
Neutral word order; the focus is more on bewaar and alles.
Ik leg mijn notitie in de map en daarin bewaar ik alles.
Moving daarin to the front emphasizes the place: in there is where I keep everything.

In main clauses, Dutch usually has verb in the second position:

Daarin bewaar ik alles.
1st position: Daarin
2nd position: bewaar (the verb)
Rest: ik alles

What does daarin mean exactly, and how is it different from erin, daar, and hierin?

Daarin is a combination of daar (there) + in (in). It means in there.

daarin – in there (refers to something not near the speaker)
hierin – in here (refers to something near the speaker)
erin – in it/inside it (neutral, often used instead of daarin or hierin in everyday speech)
daar – there (just the location, not specifically “in” something)

In this sentence:

…en bewaar alles daarin.
= …and keep everything in there (in that folder).

If you only said daar:

…en bewaar alles daar.
= …and keep everything there. (Could mean on it, at that place, not necessarily inside.)

Why is it in de map and not op de map?

In de map means inside the folder/binder, while op de map means on top of the folder/binder.

Ik leg mijn notitie in de map. – I put my note inside the folder.
Ik leg mijn notitie op de map. – I put my note on the folder (on its cover).

Because a map (folder/binder) is something you put papers into, in de map is the normal and expected preposition here.

Does map mean a geographical “map” here? I thought that was kaart.

No. In this context:

de map = a folder/binder (for papers)
de kaart = a map (geographical), or a card (birthday card, bank card, etc.)

So Ik leg mijn notitie in de map means I put my note in the folder/binder, not on a geographical map.

What is the nuance of notitie? Is it just any “note”?

Notitie usually means a written note, often short and practical:

A note you write to remember something
A brief remark in a notebook
Meeting notes (together often called notities)

Some related words:

de notitie – the note (written)
de aantekening – note, annotation (often in a book, margin notes)
het briefje – little note, slip of paper (often informal or short message)
de nota – sometimes: bill/invoice, official note, policy document

In this sentence, mijn notitie is a specific written note that you’re putting into a folder.

Why is it alles and not something like het? What does alles refer to?

Alles means everything (all things in general). In:

…en bewaar alles daarin.

alles refers to all the things/documents/notes that you keep in that folder, not just this one note.

If you used het, that would refer to one specific thing (a singular neuter noun previously mentioned), for example:

Ik heb een document. Ik bewaar het daarin.
– I have a document. I keep it in there.

But here the idea is broader: you keep everything (all your notes, documents, etc.) in that folder, so alles is appropriate.

Is bewaar a special form? How is it conjugated?

Bewaar is the ik-form (first person singular, present tense) of the verb bewaren.

Conjugation in the present tense:

ik bewaar – I keep
jij/je bewaart – you keep
hij/zij/het bewaart – he/she/it keeps
wij bewaren – we keep
jullie bewaren – you (plural) keep
zij bewaren – they keep

So in the sentence:

Ik leg mijn notitie in de map en bewaar alles daarin.
ik is still the subject of bewaar, even though it is not repeated. The form bewaar matches ik.