Jeg lægger solcremen i tasken, fordi min datter altid glemmer den, når vi går ud i varmen.

Because lægge and ligge are different verbs.

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

Here the subject jeg is actively moving solcremen, so Danish uses lægger.

Compare:

• Jeg lægger bogen på bordet. = I put the book on the table.
• Bogen ligger på bordet. = The book is lying on the table.

Also, lægger is the present tense form of lægge.

Yes, Jeg putter solcremen i tasken would also sound natural.

The difference is mainly one of nuance:

• lægger suggests laying/placing it there
• putter is a very common everyday verb meaning put / pop / stick something into something

So both work, but lægger is perfectly good Danish here.

Because they are in the definite singular form.

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

• en solcreme = a sunscreen

• solcremen = the sunscreen

• en taske = a bag

• tasken = the bag

• en varme is not common in this meaning, but varmen = the heat

This is different from English, where the is a separate word.

Because after a possessive such as min, din, hans, vores, the noun normally stays in the indefinite form.

So:

• min datter = my daughter
• not min datteren

This is the normal pattern in Danish:

• min bog = my book
• vores hus = our house
• hendes taske = her bag

Because solcreme is an en-word (common gender noun), and the matching pronoun is den.

• en solcreme → den
• et hus → det

So in this sentence:

• solcremen = the sunscreen
• den = it

Compare:

• Jeg køber en bog. Jeg læser den.
• Jeg køber et blad. Jeg læser det.

Because fordi introduces a subordinate clause, and subordinate clauses have different word order from main clauses in Danish.

In a subordinate clause, sentence adverbs like altid, ikke, ofte usually come before the finite verb:

• fordi min datter altid glemmer den

Compare that with a main clause:

• Min datter glemmer altid solcremen.

So this is a very important Danish pattern:

• main clause: subject + verb + adverb ...
• subordinate clause: subject + adverb + verb ...

Fordi means because, and it introduces the reason clause.

Here:

• Jeg lægger solcremen i tasken = main clause
• fordi min datter altid glemmer den = subordinate clause explaining why

A key thing for learners is that after fordi, Danish uses subordinate-clause word order, which is why you get altid glemmer rather than main-clause-style word order.

Here når means when / whenever.

It is used for:

• repeated situations
• general truths
• future situations

That fits this sentence, because the daughter forgets the sunscreen repeatedly: it happens when(ever) they go out in the heat.

Da is usually used for a single completed event in the past.

So:

• Når vi går ud i varmen ... = when/whenever we go out in the heat
• Da vi gik ud ... = when we went out ... (one past occasion)

Literally, it is something like go out into the heat.

Breakdown:

• går = go / are going
• ud = out
• i varmen = into the heat / out in the heat

Danish often uses i with conditions or environments:

• i solen = in the sun
• i regnen = in the rain
• i kulden = in the cold
• i varmen = in the heat

So the phrase means going outside into hot or warm weather.

Because Danish often uses the definite form in expressions about the surrounding environment or weather conditions.

So these sound natural:

• i solen = in the sun
• i skyggen = in the shade
• i regnen = in the rain
• i varmen = in the heat

Here varmen refers to the specific heat outside that they are going into. It is idiomatic Danish.

Yes. Danish uses the present tense very broadly.

In this sentence, the present tense covers two slightly different ideas:

• Jeg lægger solcremen i tasken = what I am doing now / what I do
• min datter altid glemmer den = a habitual fact

That is normal in Danish. English often chooses between I put / I’m putting and she forgets / she always forgets, but Danish often just uses the simple present.

The commas mark clause boundaries:

• main clause: Jeg lægger solcremen i tasken
• subordinate clause: fordi min datter altid glemmer den
• subordinate clause inside that: når vi går ud i varmen

So the commas help show the structure of the sentence.

You may also see Danish written without these commas before subordinate clauses, depending on which comma system is being used. The version you were shown uses a perfectly normal style with those commas included.