What are the limits and controls of lawful interaction with a girl whom a suitor wishes to marry in the future, given his inability to marry currently, and is it permissible to speak with her via social media about matters other than marriage, such as cultural discussions, knowing that the communication aims to prevent complete قطع between them?

It is permissible to speak with and look at one's fiancée for necessity only. Communication and reading some books (together) are not from the Sharia, as this may be an entry point to what is forbidden. One must repent to Allah from excessive conversations. And whoever leaves something for the sake of Allah, Allah will compensate him with something better. It is advisable to expedite the marriage contract and delay the consummation if he is unable to complete it now.