The present perfect works here. "Present" tenses can often be used in "non-past" situations, and we are talking about the state at a non-past time.
You could also use simple present: "I'll tell you if the situation changes". This slightly changes the meaning. The present perfect suggests that "I'll tell you about the situation and whether it has changed ...
What you want in this instance is to use the future continuous. The future continuous tense indicates that an action will take place in the future, and will continue to do so. It is formed using will + be + present participle.
Additionally, the expression "by the start of next week" effectively causes the reader to start counting from now, until the start ...
Now you have supplied a proper sentence, I can tell you that, no, you can't use what to be done in the same way.
I asked what to do = what I should do (an active verb).
To be done is passive, so you need to add was.
I asked what was to be done = what action needed to be taken.
The simple present is used for present events, events that happen generally ("The bus leaves at 8:45 every morning."), and as historical or narrative present. It can also be used to describe future events, a use that is called "futurate". This is similar to narrative present in using present tense to describe events at another time.
In your example,