From your question, I understood (hopefully correctly) that you want to estimate the next observation, given the observations up to now. Let $y_{1:N} = Y$ the N observations you have seen until now and let $\Theta$ be the parameters of the HMM. Then you want to infer the probability of the next observation given the already observed data, which can be expressed as:

$$ P(y_{N+1}|y_{1:N}=Y,\Theta)$$

If this is what you want, the above conditional expression is equal to : $$ P(y_{N+1}|y_{1:N}=Y,\Theta) = \dfrac{P(y_{1:N}=Y, y_{N+1}|\Theta)}{P(y_{1:N}=Y|\Theta)}$$

Note that the denominator is independent from $y_{N+1}$. So, it is:

$$ P(y_{N+1}|y_{1:N}=Y,\Theta) \propto P(y_{1:N}=Y, y_{N+1}|\Theta)$$

A brute force approach is the following:

For each of your possible observations, $y_{N+1}=Click, y_{N+1}=Scroll$ etc, calculate the likelihood of the sequences $y_{1:N+1}$. So what you need to calculate is $P(y_{N+1}=Click,y_{1:N}=Y|\Theta)$ , $P(y_{N+1}=Scroll,y_{1:N}=Y|\Theta)$, etc. for each of your possible observation sequences. Then the $y_{N+1}$ which gives the maximum likelihood can be estimated as the best guess for the next observation. Note that each of these likelihood calculations are the straightforward applications of the forward pass algorithm, which corresponds to one of the three problems of HMMs, the calculation of the likelihood of a observation sequence. You have stated this in b) in your question.

Hope this helps.

From your question, I understood (hopefully correctly) that you want to estimate the next observation, given the observations up to now. Let $y_{1:N} = Y$ the N observations you have seen until now and let $\Theta$ be the parameters of the HMM. Then you want to infer the probability of the next observation given the already observed data, which can be expressed as:

$$ P(y_{N+1}|y_{1:N}=Y,\Theta)$$

If this is what you want, the above conditional expression is equal to : $$ P(y_{N+1}|y_{1:N}=Y,\Theta) = \dfrac{P(y_{1:N}=Y, y_{N+1}|\Theta)}{P(y_{1:N}=Y|\Theta)}$$

Note that the denominator is independent from $y_{N+1}$. So, it is:

$$ P(y_{N+1}|y_{1:N}=Y,\Theta) \propto P(y_{1:N}=Y, y_{N+1}|\Theta)$$

A brute force approach is the following:

For each of your possible observations, $y_{N+1}=Click, y_{N+1}=Scroll$ etc, calculate the likelihood of the sequences $y_{1:N+1}$. So what you need to calculate is $P(y_{N+1}=Click,y_{1:N}=Y|\Theta)$ , $P(y_{N+1}=Scroll,y_{1:N}=Y|\Theta)$, etc. for each of your possible observation sequences. Then the $y_{N+1}$ which gives the maximum likelihood can be estimated as the best guess for the next observation. Note that each of these likelihood calculations is a straightforward application of the forward pass algorithm, which corresponds to one of the three problems of HMMs: The calculation of the likelihood of a observation sequence. You have stated this in b) in your question.

Hope this helps.