Learn French: Oui, je suis français.

In French, nationalities are not capitalized when they are used as adjectives to describe someone. Because this sentence describes the speaker as French, it uses the lowercase français. You only capitalize nationalities when referring to the people as a noun, like les Français (the French).

Would a woman say this sentence exactly the same way?

No. Nationalities change depending on the gender of the person they describe. A man says je suis français, but a woman adds an 'e' at the end to make it je suis française. Adding the 'e' also changes the pronunciation, making the 's' sound like a "z".

If this sentence is a reply, how did the other person ask "Are you French?"

It depends on the politeness register! If a friend or someone your own age asked, they would use the informal tu, saying Tu es français ? If a stranger, someone older, or a group of people asked, they would use the formal or plural vous, saying Vous êtes français ?

Do I pronounce the 's' at the end of suis and français?

No, the final 's' in both suis and français is silent. In French, final consonants are usually not pronounced unless they are followed by a word starting with a vowel. Since français starts with a consonant, the 's' in suis stays silent.