Yes it is entirely normal and no one should tell you to get rid of your pubic hair unless you are wanting to do it for yourself and not because of peer pressure. You can ask your mother or sister about this question.
Hair removal goes back many thousands of years and many methods have evolved over time. Some methods sound very simple and easy. The truth is that removing hair without damaging the skin and causing pain and/or infection is the problem that bedevils many people even today. Some methods are ineffective or messy, some are painful, some cause skin damage and some are wildly expensive.
Scissors can and do cut skin which is probably not what you want "down there" and they cannot cut close without nicking the skin.
Tweezing works after a fashion, but is utterly impractical if you have large areas of hair to remove or in very sensitive areas like nostrils and ears, and don't have days to waste in doing this.
NEVER use hair removal cremes "down there" as they have caustic chemicals in them like potassium thioglycolate that can burn and permanently discolor the skin as a nasty side effect from the chemicals attacking the hair protein. If the chemicals are harsh enough to break down hair, you are right to wonder if it damages the skin. Hence the manufacturers warnings on all such depilatory (hair removal) cremes and lotions.
Laser can cost thousands and requires several sessions. It doesn't work on all hair types and the FDA prohibits anyone from saying it is "permanent". It can also burn and permanently discolor the skin and is about as enjoyable as sticking hot needles into your skin.
Waxing is expensive and can cost over $1000 a year, and very painful and embarrassing Unless you like showing off your privates to a stranger, you may want to think again, apart from the cost, mess and inconvenience. Additionally the "holes" left in the skin when you pull a hair out by the root leave the skin open to infection and some waxes can cause allergic reactions.
Threading is really rolling a waxed thread across your skin to pull put hairs and so it differs little from waxing and takes a whole lot longer.
Some companies sell abrasive paper to "sand off" your unwanted hair. Unless you are a piece of furniture, the time it takes and the abrasion to the skin, makes this method a non starter.
Shaving is the best way to go but NOT with just any face shaver or a wet blade razor. It makes no difference to your skin whether the shaver you use is colored pink or blue, that is just a marketing ploy that some manufacturers use to deceive you. Unless you use a single head rotary personal shaver with a flexible foil, you may find that you still get shaver burn, lumps and ingrown hairs.
I have tried every hair removal method available and without doubt the best results for the least money is to use a good quality personal shaver like the Bare- It- All shaver and trimmer, designed for sensitive skin anywhere on the body. Personal shavers do shave as smooth as waxing if you use them dry. Water on the skin makes the skin cells swell and these cells ride up the hair shaft and conceal it from the shaver blade. Once the skin dries out the cells contract and this exposes the hair shaft that wasn't cut properly. The result is that you may feel that " shaving makes the hairs grow back quicker". Any doctor will tell you that this myth is not biologically possible.
The "Bare-It-All" web site has lots of information about shaving any hair and the various options and techniques of pubic hair removal. Just Google the name or search on "personal shaver".