When I look at options chains, there are no options for strike price $0. Not even for low-priced stocks. Not even for stocks of bankrupt companies that still have unexpired options. Why is $0 omitted?
Put option with strike price $0 would give holder right to sell stock for $0 no matter what is the stock price (put option is not exercisable if you don't own stock). Owning such put option in normal world would be punishment as stock prices don't go below $0 and at exercise you would be giving away stock for free. So such put option would always trade for $0.
Call option with strike price $0 for writer would make responsibility to sell stock to option holder for full price when assigned no matter what is the stock price. This call option would act the same as stock and I believe premiums would mirror stock price no matter what is expiration for call option.
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.)
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.