Learn Russian: У меня сильный кашель.

In English, we say we have a 'bad' cough, but Russian uses сильный (strong) to describe intense physical symptoms like pain or a cough. It is also used for intense weather conditions, like a 'strong' wind. If you translated 'bad cough' literally as плохой кашель, it would sound completely unnatural to a Russian speaker.

I originally learned that 'I have' is У меня есть. Why is есть missing here?

Russian often drops the word есть (is/exists) when describing illnesses, symptoms, or temporary physical conditions. Because you are describing a state you are experiencing rather than emphasizing the physical ownership of an object, you just say У меня followed directly by the symptom.

The word кашель ends in a soft sign. How can I tell what gender it is?

Nouns ending in a soft sign (-ь) can be either masculine or feminine, which can be tricky and usually requires memorization. However, this sentence gives you a huge clue: the adjective сильный uses the masculine ending -ый. This tells you right away that кашель is a masculine noun.

Just as a quick refresher, what grammatical case is being used for меня?