Learn Hardcore German: Der Hund ist alt, trotzdem läuft er schnell im Garten. - The dog is old, nevertheless it runs quickly in the garden.

trotzdem is a conjunctive adverb (often called a “linking adverb”) meaning nevertheless or even so. It connects two independent clauses and introduces a contrast.

Why is there a comma before trotzdem?

You’re joining two main clauses with a conjunctive adverb. In German, you must place a comma before trotzdem when it links two independent clauses: Der Hund ist alt, trotzdem läuft er schnell.

Why is the verb läuft placed before the subject er in the second clause?

German main clauses follow the V2 (verb-second) rule. Since trotzdem occupies the first position, the finite verb läuft must come next, pushing the subject er into third position.

Why is im Garten in the dative case, and why not in den Garten?

in
- dative expresses location (“where?”): in dem Garten → im Garten.
in den Garten (accusative) would express motion “into the garden.” Here the dog is running around inside the garden, so you use the dative.

Why is schnell placed before im Garten? Could it come after?

German usually orders adverbials by Time–Manner–Place (TMP).

There’s no time element here, so schnell (manner) comes before im Garten (place).
You could say läuft er im Garten schnell, but it sounds less natural and shifts the emphasis to schnell at the end.

Can trotzdem be replaced with aber or dennoch?

dennoch is a near-perfect synonym and functions identically:
Der Hund ist alt, dennoch läuft er schnell.
aber is a coordinating conjunction (“but”) and also needs a comma. With aber you would write:
Der Hund ist alt, aber er läuft schnell im Garten.
Since aber doesn’t trigger inversion, the subject-verb order stays er läuft.

Why is alt not inflected with an ending like alte?

After a linking verb (sein, werden, bleiben), adjectives are used predicatively and remain uninflected. Thus alt stays in its base form:
Der Hund ist alt.