You had a good idea, but it's not quite right. You're counting not only seats occupied by women not surrounded by women but also all seats occupied by men. (You also didn't execute the idea correctly – for corner seats you calculated a conditional probability, and for interior seats I'm not sure exactly what you calculated.)
Here are three correct ways to solve the problem:
1) Perhaps most similar to what you tried to do, we can count the number of seats occupied by a woman and adjacent to at least one man. This is the probability for the seat to be occupied by a woman minus the probability for the seat to be occupied and surrounded by women. For a corner seat, this is
$$
\frac12-\frac{\binom83}{\binom{10}5}=\frac5{18}\;.
$$
For an interior seat, it's
$$
\frac12-\frac{\binom72}{\binom{10}5}=\frac5{12}\;.
$$
The total is
$$
2\cdot\frac5{18}+8\cdot\frac5{12}=\frac{35}9\;.
$$
2) Similarly, you can count the number of seats occupied and surrounded by women, and subtract that from the total number of women. For a corner seat, this is
$$
\frac{\binom83}{\binom{10}5}=\frac29\;,
$$
for an interior seat, it's
$$
\frac{\binom72}{\binom{10}5}=\frac1{12}\;,
$$
and so the expected value is again
$$
5-\left(2\cdot\frac29+8\cdot\frac1{12}\right)=\frac{35}9\;.
$$
3) Alternatively, you could focus on the women instead of the seats. Each woman has a probability of $\frac2{10}$ of being in a corner seat, and then she has a probability of $\frac59$ of sitting next to a man, and a probability of $\frac8{10}$ of being in an interior seat, and then she has a probability of
$$
1-\frac{\binom72}{\binom94}=\frac56
$$
of sitting next to a man, again for a total of
$$
5\left(\frac2{10}\cdot\frac59+\frac8{10}\cdot\frac56\right)=\frac{35}9\;.
$$
Best Answer
Pick a particular woman W. What's the probability she's sitting next to at least one man? Well this depends on whether she's sitting at the end of the row or not, so it's easier to split into two cases.
If W is sitting on the end (probability $1/5$), there is one person next to her, and $5$ men out of the $9$ other people, so the probability it's a man is $5/9$.
If W is not sitting on the end, (probability $4/5$), there are two people next to her. The number of possible pairs of people to sit next to her is $\binom 92=36$, and all of these are equally likely. The number of pairs of other women is $\binom 42=6$, so the probability she is sitting next to at least one man is $5/6$.
So the probability W is sitting next to at least one man is $\frac 15\times\frac59+\frac45\times\frac56=\frac79$. Now the expected number of women is just this probability times the total number of women.