Nach dem Sport esse ich lieber etwas Herzhaftes statt Kuchen.

Because nach (in the sense of after) takes the dative case.

• der Sport (masculine) → dative: dem Sport
  So nach dem Sport = after (the) sport / after working out.

German normally puts the conjugated verb in position 2 in main clauses (the V2 rule). If you put something else first (here: Nach dem Sport), the subject comes after the verb:

• Ich esse nach dem Sport ... (subject first)
• Nach dem Sport esse ich ... (time phrase first → inversion)

Yes, that’s completely correct. The meaning is basically the same. The difference is mainly focus:

• Nach dem Sport esse ich ... emphasizes when.
• Ich esse nach dem Sport ... is more neutral and emphasizes I a bit more.

Because the subject is ich (first person singular):

• ich esse
• du isst
• er/sie/es isst

lieber means rather / preferably. It often sits near what it modifies, but placement is flexible:

• ... esse ich lieber etwas Herzhaftes ...
• ... esse ich etwas Herzhaftes lieber ... (possible, but usually less natural)

A very common pattern is: Subject + verb + lieber + object.

Because it’s a nominalized adjective (an adjective used as a noun). In German, nouns are capitalized:

• etwas Herzhaftes = something savory/hearty
  So Herzhaftes functions like a noun.

Because etwas is an indeclinable pronoun, and the adjective takes a strong ending as if there were no article. In neuter singular, that ending is often -es:

• etwas Gutes
• etwas Neues
• etwas Herzhaftes

Here it’s the object of essen (accusative), and neuter nominative/accusative look the same: -es.

You can, and it’s grammatically possible, but it sounds a bit more abstract/general. Most speakers prefer:

• lieber etwas Herzhaftes (very common, natural)

Without etwas, it can feel like “in general, savory things.”

Both are possible depending on style and structure:

• statt Kuchen = very common, especially as a general “instead of cake” (no article, more generic)
• statt des Kuchens = genitive, more formal/precise (“instead of the cake”)

In everyday speech, you may also hear dative:

• statt dem Kuchen (common spoken usage, prescriptively less formal)

They mean the same thing (instead of).

• statt is slightly more common/shorter.
• anstatt can sound a bit more formal or emphatic, but the difference is small.

nach has several meanings, but here it’s clearly temporal (after). Other common meanings include:

• direction: nach Berlin (to Berlin)
• according to: nach dem Gesetz (according to the law)

The case helps: nach + dative is typical for these uses.