Your translation of the text is not correct. You have to read it as two different remarks:
Kartennutzung bis 16.07.2020
This translates as:
Card usage until July 16th, 2020
The right to work is not limited in time, but the usage of the card. The reason is as you probably have guessed that the residence permit ...
First, check if your situation satisfies European directive for researchers and students, please see my another answer on this.
If you do not fit into that regulation, then I would expect that you need to apply for a residence permit in Hungary (probably, one is not allowed to stay there longer than 6 months with student residence permit of another country).
If you no longer reside in your apartment (i.e. you are giving it up), then you must deregister it within a 2 week period after leaving (earliest: one week before leaving).
Please note that when you permanently leave Germany your residence permit expires, since you have terminated your residence.
This will have no effect on your continuing employment while ...
No, through the application itsself your German residence permit will not become invalid.
Only when you permanently leave the country (i.e. you deregister your residence in Germany and leave) does the residence permit become invalid.
The corresponding paragraph is § 51 (1)(6) AufthG
(1) The residence title shall expire in the following cases:
6. if the ...