Jeg lægger mine solbriller og hendes sandaler på hylden, så de kan tørre.

Because lægge means to lay / put something down — it involves movement or placement.

• Jeg lægger solbrillerne på hylden = I put the sunglasses on the shelf.
• Solbrillerne ligger på hylden = The sunglasses are lying on the shelf.

So lægger is the right choice because the speaker is actively placing the items somewhere.

These are both possessives, but they behave differently.

• min / mit / mine change form depending on the noun.
• hans / hendes / dens / dets / vores / jeres / deres do not change.

Because solbriller is plural, you use mine.

Examples:

• min sandal = my sandal
• mit håndklæde = my towel
• mine solbriller = my sunglasses

But with hendes, the form stays the same:

• hendes sandal
• hendes sandaler

This is about the difference between ordinary and reflexive possessives in Danish.

Sin / sit / sine are used when the possessor is the subject of the clause.

In your sentence, the subject is jeg, not hun, so hendes is correct:

• Jeg lægger hendes sandaler ... = I am putting her sandals ...

If the subject were hun, then you would normally use sine:

• Hun lægger sine sandaler på hylden = She puts her own sandals on the shelf.

So:

• hendes = her, someone else's
• sine = her own, referring back to the subject

In Danish, a possessive normally replaces the article.

So you say:

• mine solbriller
• hendes sandaler

Not:

• de mine solbriller
• hendes sandalerne

This is similar to English: you say my sunglasses, not the my sunglasses.

Yes, very often. Like English sunglasses, Danish solbriller is commonly used in the plural when talking about one pair.

• solbriller = sunglasses
• et par solbriller = a pair of sunglasses

There is a singular form, solbrille, but in everyday use the plural is much more natural when referring to the object people wear.

Because hylden is the definite form: the shelf.

• en hylde = a shelf
• hylden = the shelf

In Danish, the definite article is usually added to the end of the noun.

So:

• på en hylde = on a shelf
• på hylden = on the shelf

Here, the sentence refers to a specific shelf, so hylden is natural.

Because a shelf is treated as a surface in Danish, just as in English.

So:

• på hylden = on the shelf

You would use i if something is inside something:

• i skabet = in the cupboard
• i kassen = in the box

A shelf is normally something you put things on, not in.

De refers to both mine solbriller and hendes sandaler together.

When two nouns are joined by og, Danish uses a plural pronoun:

• solbrillerne og sandalerne ... de ...

So de kan tørre means that both items can dry.

Kan is a modal verb meaning can / be able to.

So:

• så de kan tørre = so they can dry

This expresses purpose or intended result: the speaker puts them on the shelf so that drying is possible.

After a modal verb like kan, the next verb stays in the infinitive:

• kan tørre
• vil tørre
• skal tørre

If you said så de tørrer, it would sound different — more like so they are drying or so they dry, which is less natural in this context.

Because here så means something like so that, and it introduces a subordinate clause.

In a subordinate clause, Danish normally keeps the subject before the finite verb:

• så de kan tørre

If så meant then / so, starting a main clause, you would often get main-clause word order:

• Så kan de tørre = Then they can dry

So the sentence uses så de kan tørre because it expresses purpose: the items are placed on the shelf so that they can dry.