Jeg lægger min mobil ved siden af bogen.

Because lægge and ligge are different verbs in Danish:

• lægge = to lay, to put, to place something somewhere
• ligge = to lie, to be lying

In this sentence, the subject is doing the action of placing the phone, so Danish uses lægger.

• Jeg lægger min mobil ved siden af bogen = I put / am putting my phone next to the book
• Min mobil ligger ved siden af bogen = My phone is lying next to the book

A useful shortcut:

• lægge = movement / placing
• ligge = position / result

It can mean both, depending on context.

Danish present tense often covers both:

• simple present: I put
• present progressive: I am putting

So Jeg lægger min mobil ved siden af bogen could mean:

• I put my phone next to the book
• I am putting my phone next to the book

If needed, context tells you which meaning is intended.

Because mobil is a very common everyday short form for mobile phone / cell phone in Danish.

So:

• mobil = phone, cell phone
• mobiltelefon = mobile phone

Both are correct, but mobil sounds more natural in casual everyday speech.

Because bogen means the book, while en bog means a book.

In Danish, the definite article is often attached to the end of the noun:

• en bog = a book
• bogen = the book

So Danish usually does not use a separate word for the the way English does.

Here:

• ved siden af bogen = next to the book

Ved siden af means next to or beside.

It is a fixed expression:

• ved = by / at
• siden = side
• af = of

But you should learn ved siden af as one whole phrase, because the meaning is idiomatic.

Examples:

• Stolen står ved siden af bordet = The chair is next to the table
• Jeg sætter mig ved siden af dig = I sit down next to you

Because ved siden af is just the normal full Danish expression for next to.

Even though English learners may want to break it apart word by word, it is better to treat it as a single prepositional phrase. In many cases, you simply need the whole expression:

• ved siden af bogen
• ved siden af mig
• ved siden af huset

So the af is not optional here in standard Danish.

This is the normal Danish main-clause word order:

• Jeg = subject
• lægger = verb
• min mobil = object
• ved siden af bogen = place expression

So the structure is:

Subject + verb + object + place

That is very natural in Danish.

Also, Danish main clauses usually follow the verb-second principle, meaning the finite verb comes early in the clause. In this sentence, the subject comes first, so the verb comes right after it.

Yes. That is very natural.

Danish often uses:

• lægge when something is placed lying down
• sætte when something is placed standing upright

A phone is often thought of as something you lay down, so lægger works well.

Compare:

• Jeg lægger min mobil på bordet = I put my phone on the table
• Jeg sætter flasken på bordet = I put the bottle on the table

The choice often depends on how the object is positioned.

Because mobil is a common-gender noun, and with common-gender singular nouns, Danish uses min for my.

Compare:

• min mobil = my phone
• min bog = my book

But with a neuter singular noun, you use mit:

• mit hus = my house

And with plurals, you use mine:

• mine bøger = my books

So min mobil is correct because mobil is singular and common gender.

Yes, you can, but it sounds a bit different.

• lægger = neutral, standard, very natural for placing something down
• putter = put, tuck, place; often feels a bit more informal, and sometimes suggests putting something into or into position

So Jeg lægger min mobil ved siden af bogen is a very straightforward, natural choice.

Putter is possible in some contexts, but lægger is usually the safest and most idiomatic option here.

A rough guide is:

• jeg sounds a bit like yai or yigh depending on accent
• lægger sounds roughly like LEH-guh with a soft Danish g
• mobil sounds roughly like mo-BEEL
• bogen sounds roughly like BO-wen
• ved siden af sounds roughly like vel SEE-then a

A few important points:

• Danish pronunciation is often much softer than spelling suggests.
• The g in lægger is not a hard English g.
• The final -en in bogen is the definite ending meaning the.

If you are learning pronunciation, it is especially helpful to listen to native audio for words like jeg, lægger, and bogen, because they are less phonetic than they may look.