I'm in no way an expert. A year ago, I would have said they were silly BS and made up. But from my experience since, this is what I would have to say about those questions at this point in time...
1. What kind of chakras are there?
They are a mystery. Strange, swirling energy, only witness-able in an altered state of consciousness.
They each have a "quality" that is more expressive the bigger they are... ie. there are small chakras on the fingers, that don't have a discernible quality, but the larger heart chakra palpitates with states like love or for a broken heart, and the solar plexus chakra is apparent to even the most non-sensitives with deep anxiety or over-arching dominance.
2. What do chakras look like?
I have only seen them as extremely fast spinning wheels/circles, and only during OBEs. I've never witnessed cones or other descriptions I've read about. Also, for me, they have a distinctive sound like a helicopter or a strong engine reving up.
3. Where are chakras located?
Everywhere. There a huge number of chakras. Way more (in my experience) than the 3, 5 or 7 you'll read in a book. Joints, organs, and nerve bundles are the easiest places to find them.
4. What functions do chakras have?
A mystery. But it seems like they can 1) activate physical reactions like healing, hormones, feelings, etc.. and 2) have a subjective effect, such as making one more confident, comfortable, aggressive, creative, etc.. in situations depending on your focus.
5. How can you prove your answers are true?
IMHO, you can't and you never will be able to prove it to anyone. But if you experience yourself, then you don't need to prove it to anyone else.