Lunedì, se non piove, vado in biblioteca a piedi.

They usually are lowercase in Italian: lunedì, martedì, mercoledì, etc.

Here it is capitalized simply because it is the first word of the sentence. So inside a sentence you would normally write:

Vado in biblioteca lunedì, se non piove.

Also notice the written accent: lunedì ends in an accented -ì.

Italian normally uses weekdays without an article when talking about one specific Monday:

Lunedì vado in biblioteca. = On Monday / Monday I’m going to the library.

If you use the article, it often means a habitual action:

Il lunedì vado in biblioteca. = On Mondays I go to the library.

So in your sentence, lunedì refers to a particular Monday, not every Monday.

The commas separate introductory information from the main clause.

• Lunedì sets the time.
• se non piove adds a condition.
• vado in biblioteca a piedi is the main statement.

So the commas help the sentence read clearly:

Lunedì, se non piove, vado in biblioteca a piedi.

In everyday writing, punctuation can vary a little, especially after lunedì, but this version is perfectly natural and clear.

Because Italian very often uses the present indicative after se for a real future condition.

So:

se non piove = if it doesn’t rain

This is the normal, natural way to say it. Italian does not need a future form here to refer to the future.

Italian often uses the present tense for future actions when the time is already clear from context.

Here, lunedì already tells you the action is in the future, so vado sounds very natural:

Lunedì vado in biblioteca. = On Monday I’m going to the library.

You could also say andrò, but vado is extremely common for planned future actions.

Because Italian usually does not use a dummy subject like English it with weather expressions.

English says:

it is raining
if it doesn’t rain

Italian simply says:

piove = it is raining / it rains
se non piove = if it doesn’t rain

So piove works by itself.

Andare in biblioteca is the most common, idiomatic way to say to go to the library.

Italian often uses in with certain places or institutions when talking about going there as a destination or for its function:

• andare in biblioteca
• andare in banca
• andare in chiesa

Alla biblioteca is possible in some contexts, especially if you are thinking of a specific building or location, but in biblioteca is the standard choice here.

A piedi means on foot or walking.

It is a fixed expression in Italian:

• vado a piedi = I go on foot / I walk
• sono andato a piedi = I went on foot

You do not normally say con i piedi for this meaning. Just learn a piedi as the standard expression.

Because Italian usually leaves out subject pronouns when the verb already makes the subject clear.

Vado already means I go / I’m going, so io is unnecessary.

You would add io only for emphasis or contrast:

Io vado in biblioteca, lui resta a casa.
= I’m going to the library, he’s staying home.

So in your sentence, omitting io is the normal choice.

Yes. Italian word order is fairly flexible, especially with time expressions and conditional clauses.

For example, these are all natural:

Lunedì, se non piove, vado in biblioteca a piedi.
Se non piove, lunedì vado in biblioteca a piedi.
Vado in biblioteca a piedi lunedì, se non piove.

The basic meaning stays the same, but the emphasis shifts slightly depending on what comes first.

Your original version is very natural because it starts with the time, then gives the condition, then the main action.