Learn Spanish (Spain): Él es un político grande y gordo.

This relates directly to your current lesson! When grande comes after a noun, it describes literal physical size (big or large). If it comes before a singular noun, it drops the final syllable to become gran and its meaning shifts to 'great' or 'impressive' (for example, un gran político would mean 'a great politician'). Here, we are talking about his physical body.

I learned we shouldn't use un or una with professions. Why do we say un político here?

You are right that we normally drop the article when just stating someone's job, such as saying Él es político. However, the rule changes when you add descriptive adjectives to the profession. Because we are specifically describing the politician as grande y gordo, we must bring back the article un.

Why doesn't grande change to end in 'o' to match the masculine noun político?

Adjectives that end in -e in their singular form, like grande, inteligente, or triste, are gender-neutral. They have the exact same form for both masculine and feminine nouns. They only change when making them plural by adding an -s.

Since a person's weight can fluctuate, why do we use es (from ser) instead of está (from estar) for being fat?