Hunden ser mærkeligt på mig.

Because hunden is the definite form of hund.

• en hund = a dog
• hunden = the dog

In Danish, the definite article is usually added to the end of the noun as a suffix:

• en bil → bilen
• en kat → katten
• en hund → hunden

So Hunden ser mærkeligt på mig means The dog is looking at me strangely, not A dog...

Because here it is being used like an adverb, describing how the dog looks at me.

• mærkelig = strange / weird as an adjective
• mærkeligt = strangely / weirdly in this sentence

Compare:

• en mærkelig hund = a strange dog
  Here mærkelig describes the noun hund.
• Hunden ser mærkeligt på mig = The dog looks at me strangely
  Here mærkeligt describes the verb ser.

In Danish, an adjective often takes -t when used in this kind of neutral/adverb-like way.

Because Danish often uses se på when talking about looking at someone or something.

• se = see
• se på = look at

So:

• Jeg ser hunden = I see the dog
• Jeg ser på hunden = I look at the dog

In your sentence, the meaning is about the dog’s manner of looking, so ser ... på mig is natural:

• Hunden ser mærkeligt på mig = The dog looks at me strangely

Because mig is the object form of the pronoun, while jeg is the subject form.

• jeg = I
• mig = me

In this sentence, the dog is the subject, and me is the object of på:

• Hunden = subject
• på mig = at me

So Danish uses mig, just like English uses me rather than I after a preposition.

Compare:

• Jeg ser hunden = I see the dog
• Hunden ser på mig = The dog looks at me

Ser is the present tense of se.

• at se = to see / to look
• ser = see / sees / am seeing / is seeing, depending on context

Danish verbs are nice here because they do not change for person:

• jeg ser = I see
• du ser = you see
• han ser = he sees
• hunden ser = the dog sees / looks

So ser works for all subjects in the present tense.

Yes, this is normal Danish word order for a main clause.

The structure is:

• Hunden = subject
• ser = verb
• mærkeligt = adverb
• på mig = prepositional phrase

So it follows a very typical pattern: Subject + Verb + Adverb + Prepositional phrase

You could think of it as: The dog + looks + strangely + at me

Yes, you often could.

Both ser på and kigger på can relate to looking at someone, but there can be a slight difference:

• se på = look at / watch
• kigge på = look at / gaze at

In many everyday situations, both are possible.
However, ser mærkeligt på mig is a very common way to say someone is giving me a strange look.

So:

• Hunden ser mærkeligt på mig = very natural
• Hunden kigger mærkeligt på mig = also possible, a bit more focused on the act of staring/looking

It can feel like either, depending on context.

Mærkelig often means:

• strange
• odd
• weird

So mærkeligt in this sentence can be understood as:

• strangely
• oddly
• weirdly

In natural English, people might translate the whole sentence in different ways:

• The dog is looking at me strangely.
• The dog is looking at me weirdly.
• The dog is giving me a strange look.

All of those match the Danish well.

A rough learner-friendly guide is:

• mær sounds a bit like mare, but with the Danish æ
• ke is light and unstressed
• ligt is not pronounced exactly like English spelling suggests

A rough approximation might be: MAIR-kə-ligt or MAIR-kə-lee

In real speech, the ending can sound reduced, and pronunciation varies a bit by speaker and region. The important thing for learners is:

• stress is mainly near the beginning: MÆR-
• the final -t may be weak in connected speech

If you want to sound natural, it helps to listen to native audio rather than relying only on spelling.