If a lottery has 300 tickets, shouldn't I win every 300 times I play

Yes it is. The probability of winning $k$ times out of $n$ lotteries is determined from a binomial distribution:

$$P(K=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where $n=300$, $k=1$, and $p=1/300$. The answer is about $0.368494$, or $36.8\%$. The probability of winning at least once is

$$P(K \ge 1) = 1- P(K=0) = 1-\left( \frac{299}{300} \right )^{300} \approx 0.632735$$

or about $63.3\%$ chance of winning something. Not bad, but not $100\%$.

If you flip a coin twice and call it in the air both times, will you necessarily call either flip correctly? Will you necessarily call either flip incorrectly? It isn't that simple.

The very general idea here--by the Law of Large Numbers--is that, if you play this lottery "enough" times, then the fraction of times that you win (the empirical probability) will be "close to" $1/300$ (the theoretical probability). This is a fantastically imprecise notion (hence the quotes), as your tests should make clear. After $1104$ tests, you still haven't won anything, so while $0$ is arguably "close to" $1/300$, you may want your empirical probability to be closer, which means that you haven't played "enough" times, yet.

Not at all.

If your chances of winning are $1/300$ then the chances of you winning over $300$ draws is: $$1-\left(1-\frac{1}{300}\right)^{300}=0.632 \sim 63\%$$ So you still have a $37$ percent chance of not winning, even after playing $300$ draws.