You've asked for an exhaustive treatment of this martingale. I've attempted below to get it all.
There is a preliminary point to make before pursuing any properties. Let $A_n$ denote the event that the gambler wins the $n$-th hand. The way in which I interpret your rules is
$$
X_{n+1} = \frac{1}{2}(1 + X_n) ~~ \text{on }A_{n+1}\\
= \frac{1}{2} X_n ~~ \text{on }A_{n+1}^c
$$
and $P(A_{n+1} \mid \mathcal{F}_n) = X_n$. This last line is important for obvious reasons, and says literally that the gambler wins on hand $n+1$ with likelihood $X_n$, when $X_n$ is known.
- $X_n$ is a martingale.
We compute
$$
E[X_{n+1} \mid \mathcal{F}_n] = E[X_{n+1} I_{A_{n+1}} \mid \mathcal{F}_n] + E[X_{n+1} I_{A_{n+1}^c} \mid \mathcal{F}_n] \\
= E[\frac{1}{2}(1 + X_n) I_{A_{n+1}} \mid \mathcal{F}_n] + E[\frac{1}{2}X_n I_{A_{n+1}^c} \mid \mathcal{F}_n] \\
= \frac{1}{2}(1 + X_n) P(A_{n+1} \mid \mathcal{F}_n) + \frac{1}{2} X_n P(A_{n+1}^c \mid \mathcal{F}_n) \\
= \frac{1}{2}(1 + X_n) X_n + \frac{1}{2} X_n(1 - X_n) = X_n
$$
where from the second to third lines I used the rule 'taking out what is known'. This is the martingale property, as desired.
2.$E[X_{n+1}^2] = \frac{E[3 X_n^2 + X_n]}{4}$.
I'll prove this in a stronger form:
$$
E[X_{n+1}^2 \mid \mathcal{F}_n] = E[X_{n+1}^2 I_{A_n} \mid \mathcal{F}_n] + E[X_{n+1}^2 I_{A_n^c} \mid \mathcal{F}_n] \\
= E[\frac{1}{4} (1 + X_n)^2 I_{A_n} \mid \mathcal{F}_n] + E[\frac{1}{4} X_n^2 I_{A_n^c} \mid \mathcal{F}_n] \\
= \frac{1}{4} (1 + X_n)^2 X_n + \frac{1}{4} X_n^2(1 - X_n) \\
= \frac{3 X_n^2 + X_n}{4}
$$
From the second to the third lines, I took out what is known and plugged in $P(A_{n+1} \mid \mathcal{F}_n) = X_n$ in one shot. The result you want now follows from taking expectations.
3.$X_n$ converges almost surely and in $L^2$ to a random variable $Z$ for which $E[Z] = E[Z^2] = p$.
Note that your martingales are nonnegative and bounded from above by $1$. Therefore the MG convergence theorem says that $X_n$ converges in $L^2$ and almost surely. This is well-known and general and I won't discuss this theorem any further, except to point you to wikipedia.
Now it is also the case that $X_n$ is uniformly integrable, being bounded uniformly in $n$ by the constant function $1$. Therefore we use the MCT and the bounded convergence theorem to get
$$
E[X_n] \rightarrow E[Z]
$$
However $X_n$ is an MG, hence $E[X_n] = p$ for all $n$, and $E[Z] = p$ as desired.
To show that $E[Z^2] = p$ we'll have to be more clever. Iterating the identity I proved in the answer to 2, it is possible to show that for all $m < n$,
$$
E[X_n^2 \mid \mathcal{F}_m] = (\frac{3}{4})^{n-m} X_m^2 + \frac{1}{4} \left( 1 + \frac{3}{4} + \cdots + \left( \frac{3}{4}\right)^{n-m-1}\right) X_m
$$
First, let $m = 0$, and second, take the limit as $n \rightarrow \infty$. We obtain that
$$
E[Z^2] = \lim_n E[X_n^2] = 0 + \frac{p}{4} \sum_{k = 0}^{\infty} \left(\frac{3}{4} \right)^k = \frac{p}{4} \frac{4}{1} = p
$$
using the $L^2$ convergence of the MCT (or bounded convergence, depending on your mood today).
4.With $Y_n = 2 X_{n+1} - X_n$, find the law of $Y_n$, show that $Y_n \rightarrow Z$ almost surely, and compute $P(\liminf_n G_n) = p, P(\liminf_n P_n) = 1-p$.
You should realize by now that $\{Y_n = 1\} = A_{n+1}$ is precisely the event that the Gambler has won the $n+1$-th hand. To find the law, recall that
$$
P(A_{n+1} \mid \mathcal{F}_n) = X_n \\
P(A_{n+1}) = E[X_n] = p
$$
by martingality. Thus is the law of $Y_n$: a biased coin toss with likelihood $p$.
That $Y_n \rightarrow Z$ almost surely follows immediately from $X_n \rightarrow Z$.
Notice now that $Y_n$ is $\{0,1\}$-valued, and so $Z$ is almost surely $\{0,1\}$ valued as well. Realize also that $\liminf_n G_n$ is the event "always winning after a certain time" and $\liminf_n P_n$ is the event "always losing after a certain time", so we see that
$$
\liminf_n G_n \subset \{Z = 1\} \\
\liminf_n P_n \subset \{Z = 0\}
$$
Now we'll do something cheap:
$$
E[Z] = p = 1 \cdot P(Z = 1) + 0 \cdot P(Z = 0) = P(Z = 1)
$$
Now what remains is to argue that $\liminf_n G_n$ has full measure inside $\{Z = 1\}$. To see this, think about what it would mean if on a positive measure set inside $\{Z = 1\}$ we had $Y_n = 0$ infinitely often. This last part will finish item 4.
5.Are the $Y_n$ independent?
No, I don't think so. This is a bit counterintuitive, but that's life. You can compute $E[Y_0Y_1]$ by hand and see that it's not equal to $p^2$ by considering $E[Y_0Y_1 \mid \mathcal{F}_1]$ and then computing an expectation.
