Jeg ligger på sengen og hviler mig.

Because Danish usually uses a position verb where English often just uses be.

• Jeg ligger ... = I am lying ...
• Jeg sidder ... = I am sitting ...
• Jeg står ... = I am standing ...

So Jeg ligger på sengen is more natural than a literal I am on the bed-type structure if the point is that the person is in a lying position.

The dictionary form is at ligge = to lie.

In this sentence:

• jeg ligger = I am lying / I lie

This is the present tense form. Danish present tense is often very simple: you usually add -r to the infinitive.

• at ligge → ligger
• at hvile → hviler

This is a very common learner question.

• ligge = to lie / to be lying
  This describes a state or position.
• lægge = to lay / to put
  This describes a movement or action of placing something somewhere.

Compare:

• Jeg ligger på sengen. = I am lying on the bed.
• Jeg lægger mig på sengen. = I lie down on the bed / I put myself on the bed.

So in your sentence, ligger is correct because the person is already in that position.

Both can exist, but they do not mean exactly the same thing.

• på sengen = on the bed
• i sengen = in bed

So:

• Jeg ligger på sengen suggests physically lying on top of the bed.
• Jeg ligger i sengen usually means I am in bed, often under the covers or simply in the bed as a place.

English speakers often need to notice that Danish makes this distinction clearly.

Sengen is the noun seng with the definite ending attached.

• en seng = a bed
• sengen = the bed

Danish often puts the at the end of the noun instead of using a separate word like English does.

More examples:

• en stol = a chair
• stolen = the chair

So på sengen literally means on-the-bed, but in natural English we say on the bed.

at hvile sig means to rest oneself, which in natural English is usually just to rest.

So:

• hviler mig = am resting
• more literally: am resting myself

The word mig is a reflexive pronoun. It shows that the action comes back to the subject.

• jeg = I
• mig = me / myself

In this sentence, hviler mig is the natural Danish expression.

Sometimes hvile can appear without a reflexive pronoun, but in the sense of taking a rest, Danish very often uses hvile sig.

So for a learner, Jeg hviler mig is the safest and most natural choice for I am resting.

Without mig, hviler can sound more formal, literary, or can suggest a different meaning depending on context.

Because mig is the correct Danish object form of jeg.

Here are the relevant forms:

• jeg = subject form = I
• mig = object/reflexive form = me / myself
• min = my (possessive), so it would not fit here

So:

• Jeg hviler mig. = I rest myself / I am resting.

Og means and.

It connects the two actions:

• Jeg ligger på sengen = I am lying on the bed
• og hviler mig = and resting

So the whole sentence describes two connected things happening at the same time.

Yes. Both verbs are in the present tense:

• ligger = present
• hviler = present

Danish often uses the present tense where English might use either:

• I lie on the bed and rest
• I am lying on the bed and resting

In context, the English translation is usually the continuous form: I am lying on the bed and resting.

Because Danish does not build the present continuous the same way English does.

English:

• I am lying
• I am resting

Danish often just uses the simple present:

• Jeg ligger
• Jeg hviler mig

So Danish present tense can cover both:

• a general present meaning
• an action happening right now

Context tells you which one is meant.

Grammatically, yes, but in most everyday contexts the more natural English translation is:

• I am lying on the bed and resting.

That is because the Danish present tense often describes what is happening right now, even though English usually prefers the -ing form for that.

Both can relate to relaxing, but they are not exactly identical.

• hvile sig = rest, especially to recover energy
• slappe af = relax, unwind, take it easy

So:

• Jeg ligger på sengen og hviler mig = I am lying on the bed and resting
• Jeg ligger på sengen og slapper af = I am lying on the bed and relaxing

In many situations they overlap, but hvile sig has a stronger sense of resting.

A rough guide is:

• Jeg ≈ yai
• ligger ≈ LI-gər
• på ≈ paw / poh
• sengen ≈ SENG-en
• og often sounds more like å
• hviler ≈ VY-lər
• mig ≈ mai

A more natural connected pronunciation may sound something like:

yai LI-gər poh SENG-en å VY-lər mai

The exact sounds vary by speaker and region, but one very important point is that og is often reduced in normal speech.

Yes, absolutely. That would mean:

• I am in bed and resting

This may even sound more natural than på sengen in many contexts, depending on what you mean.

Use:

• på sengen if you mean on top of the bed
• i sengen if you mean in bed

Both are correct; they just describe slightly different situations.