There are two possibilities for each of the five tosses of the coin, so there are $2^5 = 32$ possible outcomes in your sample space, as you found.

What is the probability that heads never occurs twice in a row?

Your proposed answer of $13/32$ is correct.

If there are four or five heads in the sequence of five coin tosses, at least two heads must be consecutive.

If there are three heads in the sequence of five coin tosses, the only possibility is that the sequence is HTHTH.

There are $\binom{5}{2} = 10$ sequences of five coin tosses with exactly two heads, of which four have consecutive heads (since the first of these consecutive heads must appear in one of the first four positions). Hence, there are $10 - 4 = 6$ sequences of five coin tosses with exactly two heads in which no two heads are consecutive.

In each of the five sequences of coin tosses in which exactly one head appears, no two heads are consecutive.

In the only sequence of five coin tosses in which no heads appear, no two heads are consecutive.

Hence, the number of sequences of five coin tosses in which no two heads are consecutive is $0 + 0 + 1 + 6 + 5 + 1 = 13$, as you found.

What is the probability that neither heads nor tails occurs twice in a row?

Your proposed answer of $1/16$ is correct since there are only two favorable cases: HTHTH and THTHT, which gives the probability $\frac{2}{32} = \frac{1}{16}$.

What is the probability that both heads and tails occur at least twice in a row?

Your proposed answer of $15/16$ is incorrect.

Since $1 - \frac{1}{16} = \frac{15}{16}$, your answer suggests you mistakenly believed that the negation of the statement that neither heads nor tails occurs twice in a row is that both heads and tails occur at least twice in a row. The negation of the statement that neither heads nor tails occurs twice in a row is that at least two heads or at least two tails are consecutive. For instance, the sequences HHTHT and TTTTH both violate the restriction that neither heads nor tails occur twice in a row without satisfying the stronger requirement that both heads and tails occur at least twice in a row.

If both heads and tails occur at least twice in a row, then there are four possibilities:

there is a block of three consecutive heads and a block of two consecutive tails

there is a block of three consecutive tails and a block of two consecutive heads

there is a block of two consecutive heads and a single head that are separated by a block of two consecutive tails

there is a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads

A block of three consecutive heads and a block of two consecutive tails can occur in two ways, HHHTT and TTHHH. By symmetry, a block of three consecutive tails and two consecutive heads can occur in two ways. A block of two consecutive heads and a single head that are separated by a block of two consecutive tails can occur in two ways, HHTTH and HTTHH. By symmetry, a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads can occur in two ways. Hence, there are $2 + 2 + 2 + 2 = 8$ favorable cases, giving a probability of $\frac{8}{32} = \frac{1}{4}$.