Ich bin erkältet, deshalb bleibe ich heute zu Hause.

Because German commonly expresses this idea with sein + erkältet: Ich bin erkältet.

Here, erkältet means having a cold / suffering from a cold. German does also allow a noun phrase:

Ich habe eine Erkältung. = I have a cold.

But Ich habe erkältet is not correct. In this sentence, Ich bin erkältet is the natural everyday way to say it.

Erkältet is originally the past participle of sich erkälten (to catch a cold), but in sentences like this it functions like an adjective.

So in Ich bin erkältet, it behaves much like:

• Ich bin müde. = I am tired.
• Ich bin krank. = I am sick.

That is why it goes with bin.

Because German main clauses follow the verb-second rule.

In a normal clause:

• Ich bleibe heute zu Hause.

If you put deshalb at the beginning, it takes the first position, so the finite verb must still stay in second position:

• Deshalb bleibe ich heute zu Hause.

So the word order changes:

• first position: deshalb
• second position: bleibe
• then subject: ich

This is a very common pattern in German.

Deshalb means therefore, for that reason, or that’s why.

It connects the second clause to the first one logically:

• Ich bin erkältet → reason
• deshalb bleibe ich heute zu Hause → result

So the sentence structure is basically: I have a cold; therefore, I’m staying home today.

Yes, but the structure changes.

With deshalb, you have two main clauses:

• Ich bin erkältet, deshalb bleibe ich heute zu Hause.

With weil, the reason goes into a subordinate clause, and the verb moves to the end:

• Ich bleibe heute zu Hause, weil ich erkältet bin.

So the meaning is very similar, but the grammar is different:

• deshalb → main clause word order
• weil → subordinate clause, verb at the end

Because Ich bin erkältet and deshalb bleibe ich heute zu Hause are two separate main clauses.

The comma helps separate them clearly. You could also write this with a semicolon or even as two sentences:

• Ich bin erkältet; deshalb bleibe ich heute zu Hause.
• Ich bin erkältet. Deshalb bleibe ich heute zu Hause.

So the comma marks a break between two complete thoughts.

Because bleiben expresses location, not movement.

• zu Hause = at home
• nach Hause = homeward / to home

So:

• Ich bleibe zu Hause. = I stay at home.
• Ich gehe nach Hause. = I go home.

A good shortcut is:

• Where? → zu Hause
• Where to? → nach Hause

Yes, heute can move, because German word order inside the middle of the sentence is somewhat flexible.

The version here is very natural:

• Deshalb bleibe ich heute zu Hause.

But you could also say:

• Deshalb bleibe ich zu Hause heute.
  This is grammatically possible, but less natural in most contexts.

German often prefers a general order like: subject + time + manner + place

So ich heute zu Hause sounds very normal:

• ich = subject
• heute = time
• zu Hause = place

This expression is historically based on the noun Haus, so Hause is capitalized.

In traditional and very common spelling, you write:

• zu Hause
• nach Hause

You may also sometimes see zuhause in modern usage, especially when it functions more like an adverb, but zu Hause is always a safe and very standard choice for learners.

Yes. A few natural alternatives are:

• Ich habe eine Erkältung, deshalb bleibe ich heute zu Hause.
• Ich bin erkältet, deswegen bleibe ich heute zu Hause.
• Ich bleibe heute zu Hause, weil ich erkältet bin.

These all express nearly the same idea, but they differ slightly in style and structure:

• erkältet sein is very common and conversational
• eine Erkältung haben is also very normal
• deshalb / deswegen are result connectors
• weil introduces the reason in a subordinate clause