Stocks - at least, blue-chip stocks - are usually thickly traded. The full set of shares outstanding does get bought and sold, at prices that usually remain roughly reflective of the market price (e.g. Dell being taken private, for a recent example). There are stocks that trade much more thinly, and this kind of absurdity can happen there, e.g. Bigfoot Project: https://www.bloomberg.com/view/articles/2016-08-18/bigfoot-r... .
So there's no clear bright line between "legitimate market cap" and "silly market cap". But there is a quantitative difference, and it's worth thinking about how deep the order book is when you see these "market cap" calculations being thrown arond.
Stocks represent ownership of an actual company that produces things and sells them, creating value. The price of the stock reflects an estimate of the future profits, the true value. If you buy all AAPL shares, Apple's future profits are yours. If some people (for whatever reason) sell AAPL while the estimate of future profits is not affected, price will drops somewhat and then value investors will swoop in to pick up the cheap shares. Others might disagree with the assessment, but future profits would vindicate the value investors. In other words, true value is realised and revealed over time. Thus, you have feedback loops that should keep the share price somewhat anchored to true value.
However, with Dogecoin, there is no intrinsic value, but solely what people assign to it. If people sell Dogecoin, that is ipso facto evidence that it's worth less, which can set of vicious cycles, run-away feedback loops, both up (bubbles) or down (crash). There is no true value. It's different from stocks.
In theory no, but in practice stocks on major markets will have a deeper book and also a "market maker" that would act as a counterparty for large trades.