Why would a 16 year old cat lose weight and fur with no appearance of sores?

My cat is 16 years old. She has generally been a very healthy cat, with a good appetite for food and activity, and somehow managed to avoid getting fleas all but twice despite being an indoor/outdoor cat.

She had fleas for the second time about a month ago. I bought Sergeant's flea spray for cats to use on the furniture and put a flea collar of the same brand on her.

About two weeks ago I noticed she had (seemingly very suddenly) lost a lot of weight. I could probably wrap my hand around her if I wanted to and if she'd let me. She's also begun scratching or biting me if I touch her torso.

Today I noticed she seems to be going bald. The fur on both sides of her torso is short and thinned out, so you can plainly see her skin on each side from behind.

There are some hard yellow bumps on her spine near her tail, but there is still plenty of fur there. There are no sores or lesions of any kind on the balding areas.

I started worrying when she lost weight; now I'm just plain scared.