Does the data section get mapped into the process' memory, or does the
program read it from the disk?
Yes, everything before execution by the dynamic loader of operating systems either Windows or Linux must be mapped into memory.
If it does get mapped into its memory, how can the process acquire the
offset of the section? ( And other sections )
PE file has a well-defined structure which loader use that information and also parse that information to acquire the relative virtual address of sections around ImageBase. Also, if ASLR - Address randomization feature - was activated on the system, the loader has to use relocation information to resolve those offsets.
Is there any way the get the entry point of a process that has already
been mapped into the memory, without touching the file on disk?
NOPE, the loader of the operating system for calculation of OEP uses ImageBase + EntryPoint member values of the optional header structure and in some particular places when Address randomization is enabled, It uses relocation table to resolve all addresses. So we can't do anything without parsing of PE file on the disk.