Lyfjaskömmtunin byrjar á mánudag.

Because Icelandic very often forms compound nouns as a single word.

Lyfjaskömmtunin can be broken down like this:

• lyfja- = related to lyf (medicine, drug)
• skömmtun = dosing, dosage
• -in = the definite article, meaning the

So the whole word is literally something like the medicine-dosing, i.e. the dosage / the dosing regimen.

It is the definite article: the.

In Icelandic, the is usually attached to the end of the noun instead of being a separate word.

So:

• lyfjaskömmtun = medication dosing / dosage
• lyfjaskömmtunin = the medication dosing / the dosage

In Icelandic compounds, the first part is often a special compound form, very often based on a genitive plural form.

Here:

• lyf = medicine, drug
• lyfja- = roughly of medicines / medication-related

You do not need to translate that part literally every time, but it helps explain why the form is lyfja-.

It is feminine.

The gender of a compound noun is determined by its last/main element, and here that is skömmtun, which is feminine.

That is why the definite form is -in here:

• skömmtun
• skömmtunin

So even though the word begins with lyfja-, the important part for gender is skömmtun.

Byrjar is the 3rd person singular present tense of að byrja (to begin / to start).

The subject is Lyfjaskömmtunin, which is singular, so the verb must also be singular:

• ég byrja = I begin
• þú byrjar = you begin
• hann/hún/það byrjar = he/she/it begins

Here the subject is like it, so:

• Lyfjaskömmtunin byrjar = The dosage starts

Because að is used with the infinitive, the dictionary form of the verb:

• að byrja = to begin

But in the sentence, the verb is conjugated:

• byrjar = begins / starts

So:

• að byrja = to begin
• byrjar = begins

You only use að with the infinitive, not with a normal finite verb in a sentence like this.

Yes. Very often.

Icelandic, like English, often uses the present tense for a scheduled or expected future event, especially when there is a time expression that makes the future clear.

So:

• Lyfjaskömmtunin byrjar á mánudag.

is literally present tense, but it naturally means:

• The dosage starts on Monday or
• The dosage will start on Monday

The phrase á mánudag makes the timing clear.

Because á is the normal preposition for on with days of the week in Icelandic time expressions.

So:

• á mánudag = on Monday
• á þriðjudag = on Tuesday
• á föstudag = on Friday

Here á is not about physical location; it is just the standard way to say on with weekdays.

Because mánudagur is the nominative form, but after á in this kind of time expression Icelandic uses mánudag, which is the accusative singular.

So:

• mánudagur = nominative, the basic dictionary form
• á mánudag = accusative after á in the expression on Monday

This is very common with weekdays.

Yes, absolutely.

That just puts the time expression first for emphasis:

• Lyfjaskömmtunin byrjar á mánudag.
• Á mánudag byrjar lyfjaskömmtunin.

Both are correct.

This also shows an important Icelandic pattern: in a main clause, the finite verb usually stays in the second position. So if Á mánudag comes first, byrjar still comes second.

Yes, you can hear both.

• á mánudag = a neutral on Monday
• á mánudaginn = also common, often a bit more specific or tied to a particular Monday in context

For a learner, á mánudag is a very good standard form to know and use. If you later hear á mánudaginn, do not be surprised—it is also natural Icelandic.