Vtyjkyuk

I've often heard the phrase what do we have here to mean what do you have. And also, recently, I've heard a teacher ask one of his students struggling with an assignment: do we have a problem?, as in do you have a problem?

I simply want to know how exactly such a rearrangement of pronoun usage became a thing and maybe if other languages do something similar. Thanks.

This is often called the "patronizing we", among other names (see this answer of mine for more details on its names). According to the Oxford English Dictionary's page for "we" (pron., n., and adj.), it first appeared in 1702:

Well, old Acquaintance, we are going to be Married then?
_{False Friend}

In comparison, the "royal we" is much older, dating back to Old English.

According to one paper about healthcare (which calls it "inclusive we"), this form can "create a sense of shared connection and shared responsibility", so that's why I assume it came into being. Further into the paper it says however that "the impression may just as likely be paternalistic and
condescending", which is probably why it's also called the "patronizing we".

Nice observation. There are four ways in which we can be used.

In your example.

what do we have here, it can be anything from given four options. It depends on the context.

Whereas,

do we have a problem? it is a condescending "we".

There are four ways:

1. Condescending we (Oxford Dictionaries)

2. 
   Nosism: A nosism is the use of 'we' to refer to oneself. Nosism, from the Latin nos, "we", is the practice of using the pronoun "we" to refer to oneself when expressing a personal opinion.

a). Royal "we"

Example:"By the Grace of God, We, Alexander I, Emperor and Autocrat of All the Russias ...."

b). Editorial "we"

Example: We are sorry to publish this news article.

c). Author's "we" or pluralis modestiae

Example: By adding three and five, we obtain eight.

3. 
   an inclusive we: inclusive "we" specifically includes the addressee (that is, one of the words for "we" means "you and I and possibly others")
   
   

   
   

   Example: We can all go to the villain's lair today.

   
   

   
   


4. 
   

   an exclusive we: exclusive "we" specifically excludes the addressee (that is, another word for "we" means "he/she/they and I, but not you"), regardless of who else may be involved.

   
   
   

   
   

   Example:We mean to stop your evil plans!

   
   

   
   
   

   Wikipedia states that second person usage of "we" (you).

   
   
   

   We (as second person): Clusivity in the second person is conceptually simple but nonetheless if it exists is extremely rare, unlike clusivity in the first. Hypothetical second-person clusivity would be the distinction between "you and you (and you and you ... all present)" and "you and someone else whom I am not addressing currently."

   
   

There is a beautiful paper on clusivity:

Simon, Horst J. Only you? Philological investigations into the alleged inclusive-exclusive distinction in the second person plural, in: Elena Filimonova (ed.): Clusivity: Typology and case studies of the inclusive-exclusive distinction.

Wikipedia link: Clusivity