Why are there no options with $0 strike price?

Because there is no market for it. Nobody wants to pay a commission for getting a worthless stock, neither for selling it.
Same reason why there is no market for used toilet paper — nobody would want to buy it.

As user389238 says, a zero-strike put is worthless, so that part is easy.

And a zero-strike American call is very similar to the stock itself (because it can be exchanged for the stock). The one difference is that the call doesn't pay a dividend when the stock does, but the call does drop in value like the stock does when going ex-dividend. So no one would hold the call across a dividend -- its open interest would disappear and then reappear after. (This already happens with low nonzero call strikes during dividends.)

On the other hand, a zero-strike European call would be pre-discounted for all dividends to be paid by expiration; it would be more like a future.

Bottom line, zero-strike calls would be similar to existing instruments and lack the distinctive "optionality" (dependence on implied volatility) that motivates options as an asset class.