Der Kleber ist fast leer, deshalb kauft mein Bruder morgen neuen Kleber für das Schulprojekt.

In German, all nouns are capitalized. So:

• der Kleber
• das Schulprojekt
• der Bruder

This is different from English, where common nouns are normally lowercase.

Because Kleber is a masculine noun in German, so its basic article is der.

In the nominative case, masculine nouns usually take der:

• der Kleber
• der Bruder

That is why the sentence starts with Der Kleber ist ...

Here, leer is a predicate adjective. After verbs like sein, werden, and bleiben, adjectives usually do not take endings.

So you get:

• Der Kleber ist leer.
• Der Kleber ist fast leer.

Compare that with an adjective directly before a noun, where an ending is needed:

• neuer Kleber
• neuen Kleber

So:

• ist leer = no adjective ending
• neuen Kleber = adjective ending required

Because the sentence contains two main clauses:

1. Der Kleber ist fast leer
2. deshalb kauft mein Bruder morgen neuen Kleber für das Schulprojekt

In German, when two independent main clauses are joined like this, a comma is commonly used.
Also, deshalb is not a subordinating conjunction like weil. It is an adverb meaning therefore / for that reason, so it starts a new main clause.

Because German main clauses follow the verb-second rule, often called V2.

That means the finite verb must be in the second position of the clause.

In the second clause, deshalb comes first, so the verb must come next:

• Deshalb kauft mein Bruder ...

This is very normal in German:

• Heute kommt er.
• Morgen arbeitet sie.
• Deshalb kauft mein Bruder ...

So deshalb takes the first position, and kauft must be second.

Again, this is because of the verb-second rule.

English often keeps subject + verb order:

• Therefore my brother buys ...

But German does not do that if another element is placed first. If deshalb is in first position, the verb must still stay second:

• Deshalb kauft mein Bruder ...

So the subject mein Bruder comes after the verb here.

Morgen is a time adverb, meaning tomorrow.

In this sentence it appears in the middle field:

• deshalb kauft mein Bruder morgen neuen Kleber ...

Yes, it can move, depending on emphasis. For example:

• Morgen kauft mein Bruder neuen Kleber ...
• Deshalb kauft morgen mein Bruder neuen Kleber ...
  This is possible, but less neutral.

The version in your sentence is a very natural, standard word order.

Because Kleber is the direct object of kauft, so it is in the accusative case.

The verb kaufen takes an accusative object:

• Er kauft Kleber.

Since Kleber is masculine, the adjective has to show the masculine accusative form:

• neuen Kleber

So:

• nominative: neuer Kleber
• accusative: neuen Kleber

That is why the sentence uses neuen Kleber.

Here Kleber is being used a bit like a material / uncountable noun, similar to milk or paper in English. So German can say:

• Er kauft neuen Kleber.

This means he is buying some new glue / new adhesive.

If you wanted to emphasize one item, such as one glue stick or one container, you could also say:

• Er kauft einen neuen Kleber.

So both are possible, but they are slightly different in feel:

• neuen Kleber = glue as a substance or general supply
• einen neuen Kleber = one specific glue item

The preposition für always takes the accusative case.

So:

• für das Schulprojekt
• für den Bruder
• für die Schule

Since Schulprojekt is neuter (das Schulprojekt), its accusative form is also das Schulprojekt.

So in this sentence:

• für = requires accusative
• das Schulprojekt = accusative singular neuter

German very often builds compound nouns by joining words together into a single word.

So:

• Schule
  • Projekt → Schulprojekt

This is extremely common in German:

• Hausaufgabe
• Schulbuch
• Sommerferien

English often uses separate words, but German usually prefers one combined noun.

Yes, but the structure would change.

Deshalb means therefore / for that reason and introduces a main clause:

• Der Kleber ist fast leer, deshalb kauft mein Bruder morgen neuen Kleber.

If you use weil, you create a subordinate clause, and the verb goes to the end:

• Mein Bruder kauft morgen neuen Kleber, weil der Kleber fast leer ist.

So the difference is:

• deshalb = result, main clause word order, verb second
• weil = because, subordinate clause, verb at the end

Both are natural, but they are built differently.

Because mein Bruder is the subject of the verb kauft, so it must be in the nominative:

• mein Bruder kauft ...

The later Kleber is the thing being bought, so it is the direct object and must be in the accusative:

• ... kauft neuen Kleber

So the sentence has:

• mein Bruder = subject, nominative
• neuen Kleber = direct object, accusative

That subject/object distinction is one of the main reasons German changes articles and adjective endings.