For your four example questions, it looks rather easy, if I am reading this correctly.
The first asks for the probability the individual has tuberculosis or cancer given not having tuberculosis and not having cancer. That should be 0.
The second, third and fourth ask for the probability the individual has tuberculosis or cancer given some combination of having them. That should be 1 each time.
I suspect that these are not quite what you are trying to ask. For example the proportion of individuals who have both tuberculosis and cancer might be $0.0104 + 0.055 - 0.064828 = 0.000572$. This is also suggested if the lack of an arrow between tuberculosis and cancer means they are independent and $0.0104 \times 0.055 = 0.000572$.
But there is no such clarity for example with the relationship between a visit to Asia and tuberculosis. This is not independent as there is an arrow, but more individuals have tuberculosis than visited Asia. Are we supposed to assume that everyone who visited Asia now has tuberculosis? It does not say that.