Expected Ratio of Coin Flips

Your bullet points amount to saying that you're going to flip the coin until the number of heads exceeds the number of tails. Suppose that this happens on the $n$-th flip; then after $n-1$ flips you must have had equal numbers of heads and tails, so $n=2m+1$ for some $m$, you now have $m+1$ heads, and the ratio of heads to flips is $\frac{m+1}{2m+1}$. If $p_n$ is the probability of stopping after the $(2n+1)$-st flip, the expected ratio of heads to flips is $$\sum_{n\ge 0}\frac{p_n(n+1)}{2n+1}\;.$$ Thus, the first step is to determine the $p_n$.

Clearly $p_0=\frac12$: we stop after $1$ toss if and only if we get a head. If we stop after $2n+1$ tosses, where $n>0$, the last toss must be a head, half of the first $2n$ tosses must be heads, and for $k=1,2,\dots,2n$ the first $k$ tosses must not include more heads than tails. The problem of counting such sequences is well-known: these are Dyck words of length $2n$, and there are $C_n$ of them, where $$C_n=\frac1{n+1}\binom{2n}n$$ is the $n$-th Catalan number. Each of those $C_n$ sequences occurs with probability $\left(\frac12\right)^{2n}$, and each is followed by a head with probability $\frac12$, so $$p_n=C_n\left(\frac12\right)^{2n}\cdot\frac12\;,$$ and the expected ratio is $$\frac12\sum_{n\ge 0}C_n\left(\frac12\right)^{2n}\frac{n+1}{2n+1}=\frac12\sum_{n\ge 0}\frac1{4^n(2n+1)}\binom{2n}n\;.$$

Very conveniently, the Taylor series for $\arcsin x$ is $$\arcsin x=\sum_{n\ge 0}\frac1{4^n(2n+1)}\binom{2n}nx^{n+1}\;,$$ valid for $|x|\le 1$, so the expected ratio is $$\frac12\sum_{n\ge 0}\frac1{4^n(2n+1)}\binom{2n}n=\frac12\arcsin 1\approx 0.7854\;.$$

Added: I should emphasize that this calculation applies only to the stated strategy. As others have noted, that strategy is known not to be optimal, though it's quite a good one, especially for being so simple.