Ich lege den Merkzettel neben das Wörterbuch, damit ich ihn nicht vergesse.

Because both nouns are neuter or masculine/feminine with different genders and cases:

• der Merkzettel = masculine
• das Wörterbuch = neuter

In this sentence, both are in the accusative:

• den Merkzettel: accusative of der Merkzettel
• das Wörterbuch: accusative of das Wörterbuch
  (for neuter, nominative and accusative look the same)

So den shows masculine accusative, while das is neuter and stays das.

Neben is a two-way preposition. That means it can take:

• accusative for movement toward a position
• dative for location in a position

Here, the sentence describes putting the note next to the dictionary, so there is movement/change of position:

• neben das Wörterbuch = next to the dictionary (placing it there)

If the note were already there and you were just describing its location, you would use dative:

• Der Merkzettel liegt neben dem Wörterbuch.
  = The note is lying next to the dictionary.

German often uses different verbs depending on how something is placed.

• legen = to lay something down, put something into a lying/horizontal position
• stellen = to stand something upright
• setzen = to set someone/something into a sitting position

A Merkzettel is something you normally lay down, so legen is the natural choice.

That is why:

• Ich lege den Merkzettel ... = I put/lay the note ...

Damit means so that or in order that. It introduces a purpose clause.

So:

• ..., damit ich ihn nicht vergesse.
  = ..., so that I don’t forget it.

It explains why the speaker puts the note next to the dictionary.

Because damit introduces a subordinate clause, and in German subordinate clauses usually send the finite verb to the end.

Main clause:

• Ich lege den Merkzettel neben das Wörterbuch

Subordinate clause:

• damit ich ihn nicht vergesse

That final verb position is a very important German pattern.

Ihn refers to den Merkzettel.

Since Merkzettel is masculine (der Merkzettel), the matching pronoun is:

• nominative: er
• accusative: ihn
• dative: ihm

In ich vergesse ihn, the pronoun is the direct object, so German uses the accusative form ihn.

The position of nicht can be tricky in German. Here, nicht negates the verb idea forget him/it, so ihn nicht vergesse sounds natural.

• damit ich ihn nicht vergesse = so that I do not forget it

If you said nicht ihn, that would sound more like a contrast:

• damit ich nicht ihn vergesse, sondern etwas anderes
  = so that I don’t forget him/it, but something else

So in the original sentence, ihn nicht vergesse is the normal wording.

Yes. Because the subject is the same in both parts of the sentence (ich), German often prefers um ... zu for purpose:

• Ich lege den Merkzettel neben das Wörterbuch, um ihn nicht zu vergessen.

This means the same thing.

A useful rule:

• use um ... zu when the subject is the same
• use damit when the subject is different, or when you want a full clause

So both are correct here, but um ihn nicht zu vergessen is also very natural.

Because German uses a comma to separate a subordinate clause from the main clause.

Since damit ich ihn nicht vergesse is a subordinate clause, it must be set off with a comma:

• Ich lege den Merkzettel neben das Wörterbuch, damit ich ihn nicht vergesse.

This comma is required in standard German.

In German, all nouns are capitalized, not just proper names.

So:

• ich = not capitalized
• lege = not capitalized
• Merkzettel = capitalized because it is a noun
• Wörterbuch = capitalized because it is a noun

This is one of the most noticeable spelling differences from English.

The main clause follows the normal German pattern where the finite verb is in second position:

• Ich = subject
• lege = finite verb
• den Merkzettel = direct object
• neben das Wörterbuch = prepositional phrase

So the structure is roughly:

• Subject + Verb + Object + Other information

German is flexible, but the finite verb usually stays in second position in a main clause.

In this sentence, yes, it would change the meaning and sound wrong with lege.

• Ich lege den Merkzettel neben das Wörterbuch
  = I place the note next to the dictionary

Using dem would suggest location rather than movement:

• Der Merkzettel liegt neben dem Wörterbuch
  = The note is next to the dictionary

So with legen and the idea of placing something, accusative is the expected choice: neben das Wörterbuch.