Your attempt is excellent, however I think it is essential to point some things out:
1) Your proof of the first part is correct, but if you are a fan of clear writing, you could do with a little rephrasing and chopping as follows: Let $\epsilon > 0$. Then, consider a subsequence $\{p_{n_k}\}$. Note that for this $\epsilon$, there is a natural number $N$ such that $n > N \implies |p_n-p| < \epsilon$. Now, because $n_k \geq k$ for every $k$, it follows that for some $K$, $n_k > N$ for all $k > K$. Hence, considering this $K$ as your new $N$, conclude that $|p_{n_k}-p| < \epsilon$ for all $k > K$. Thus, $p_{n_k}$ converges to $p$.
2) A sequence certainly can have an infinite number of subsequences. For example, if you have a sequence $1,2,3,....$, then $2,3,...$ is a subsequence, so is $3,4,...$, so is $4,5,...$ etc. Hence it is possible. However, if your sequence is constant, for example, then you cannot have infinitely many subsequences, simply because they are all the same.
2.5) This is the contentious part. There is no finiteness condition on $I$, which creates the simple problem for the existence of the maximum. For example, suppose $N=1$ for $i_1$,$N=2$ for $i_2$, $N=3$for $i_3$ and so on, then $N$ doesn't have a maximum if $I$ is infinite, which it is most of the time. Hence, the argument breaks down here.
3) That's true,but then the argument has already broken down, so it is of no significance.
Now, how you do the second part may come as a bit of a farce. The reason is this:
Suppose I have a convergent sequence, $a_1,a_2,...$ that converges to $a$. I thought of the first number in my mind, called it $a_0$, and considered the sequence $a_0,a_1,a_2,...$,which I renumber and call $b_1,b_2,...$. Well, this sequence is also convergent!
Proof: Suppose you are given $\epsilon > 0$. Then, there exists $N \in \mathbb{N}$ such that $n > N \implies |a_n-a| <\epsilon$. But now, obviously, $b_{n+1}=a_{n}$, so therefore considering the number $N+1$, $n>N+1 \implies n-1>N \implies |a_{n-1}-a| <\epsilon \implies |b_n-a| <\epsilon$, so $N+1$ works for this $\epsilon$ and $b_n$. Hence, it follows that $b_n$ is convergent.
Now, all you need is this little trick: From your sequence $\{ a_n\}$, remove the first element $\{a_1\}$. Whatever is remaining is a subsequence of $\{a_n \}$, which is given to be convergent. But then, put back $a_1$, and then the resulting sequence, which is $\{ a_n\}$ , is convergent by the fact above! Hence, we are done.
This result is somewhat odd, because we did not seem to use the whole strength of the statement "all subsequences converge". But that's okay, we've got your result!
Best Answer
Let $x_k=1$ if $k=2^i-1$ for some $i\in\mathbb{N}$, and $x_k=0$ otherwise. Also, let $k_i=2^i$. Then $x_{k_i+p}$ converges to $0$ for any $p\in\mathbb{N}$, because this subsequence has only finitely many nonzero terms. But $x_k$ clearly doesn't converge.