Learn English – correct: “will be not” or “will not be”?

You certainly don't want "future tense" for "see". Many if not most native speakers wouldn't use it for "depend" either...

It depends on what I see.

Will you need... and Will you be needing... differ in that the second of these is a less direct and hence more polite way to phrase the question.

Swan in Practical English Usage (p196) explains it well:

The tense can be used to make polite enquiries about people's plans. (By using the future progressive to ask 'What have you already decided?', the speaker shows that he/she does not want to influence the listener's intentions.)

Compare:

'Will you be staying in this evening?' (very polite enquiry suggesting I simply want to know your plans')

'Are you going to stay in this evening?' (pressing for a decision).

This usage is possible with verbs that do not normally have progressive forms.

Will you be wanting lunch tomorrow?

Need is another of the verbs that are not normally used in the progressive form, but can be in contexts such as those exemplified above.