Learn Hardcore Swedish: Han har ont i ryggen idag.

You do not say är ont for this meaning. Ont is treated more like “pain” here, and ha ont is a fixed expression meaning “to be in pain” or “to hurt (in some body part)”.

Why is it ryggen (the back) and not hans rygg (his back) or sin rygg (his own back)?

With body parts, Swedish often uses the definite form (the + noun) instead of a possessive pronoun when it’s clear whose body part is meant.

Han har ont i ryggen.
Literally: He has pain in the back → means “his back”.
Hon bröt armen. = She broke the arm → means “her arm.”

You would only use hans rygg / sin rygg if you need to emphasize whose back, for example in a contrast:

Han har ont i ryggen, inte i hennes rygg.
He has pain in his back, not in her back.

But in normal sentences about someone’s own body, the definite form alone is standard.

What exactly does ryggen mean in terms of grammar?

Rygg is a common gender noun: en rygg (a back).

Its basic forms are:

Indefinite singular: en rygg (a back)
Definite singular: ryggen (the back)
Indefinite plural: ryggar (backs)
Definite plural: ryggarna (the backs)

In Han har ont i ryggen, ryggen is:

singular
definite
used as the object of the preposition i.

Why is the preposition i used (ont i ryggen) and not på?

For pain in a body part, Swedish almost always uses i:

ont i ryggen – pain in the back
ont i huvudet – pain in the head
ont i magen – pain in the stomach
ont i knät – pain in the knee

You would not normally say ont på ryggen for simple back pain.
På can appear in other contexts, for example:

Ett sår på ryggen – a wound on the back.

So: ha ont i + body part is the pattern to memorize.

Can I say Hans rygg gör ont instead of Han har ont i ryggen?

Hans rygg gör ont is grammatically correct but much less common and sounds a bit unusual or bookish in everyday speech.

The standard, natural way to talk about pain is still:

Han har ont i ryggen.

You can use gör ont in some phrases:

Det gör ont. – It hurts.
Det gör ont när jag går. – It hurts when I walk.

But when naming a body part, har ont i + body part is by far the most common.

How would I say “My back hurts” and “Do you have back pain?” in Swedish?

Use the same pattern with different pronouns:

My back hurts.
Jag har ont i ryggen.
Do you have back pain?
Har du ont i ryggen?

Note that even for “my back” or “your back”, Swedish still uses ryggen (“the back”), not a possessive pronoun. Context tells you whose back it is.

Where can I put idag in the sentence? Is the word order flexible?

Yes, idag (today) is fairly flexible. All of these are correct:

Han har ont i ryggen idag.
Idag har han ont i ryggen.
Han har idag ont i ryggen. (more formal/written, less common in speech)

The most neutral in everyday speech is usually:

Han har ont i ryggen idag.
or
Idag har han ont i ryggen.

The meaning doesn’t really change; it’s just a question of emphasis and style.

What’s the difference between idag and i dag?

They are spelling variants of the same word and mean exactly the same thing: today.

idag – more modern, very common.
i dag – older spelling, still correct and often seen in formal writing.

In speech, there is no difference at all. In your sentence, both are fine:

Han har ont i ryggen idag.
Han har ont i ryggen i dag.

How would I say the same idea in the past or future, like “He had back pain yesterday” or “He will have back pain tomorrow”?

Keep the same structure and just change the tense and the time word:

He had back pain yesterday.
Han hade ont i ryggen igår.
He will have back pain tomorrow.
Commonly in Swedish you’d say:
Han kommer att ha ont i ryggen imorgon.
or in more casual speech:
Han kommer ha ont i ryggen imorgon.

The core pattern ha/har/hade ont i + body part stays the same.

Is Han har ont i ryggen idag formal, informal, or neutral?

It is completely neutral Swedish:

fine in everyday conversation
fine in writing
fine at the doctor’s office:
- Jag har ont i ryggen idag. – I have back pain today.

There is nothing slangy or unusually formal about it; it’s the standard way to express this idea.