As reusing a widely used algebraic group for Diffie-Hellman key exchanges might lead to far easier third-party key discovery through precomputation for that specific group, I would like to know what can possibly go wrong when generating custom groups for use as Diffie-Hellman parameters.

Here is what I currently believe to be true about the requirements. Please correct me or explain further if this is wrong or not detailed enough:

An algebraic field is needed for the Diffie-Hellman key exchange to work.

Rings without zero divisors are also integral domains. Integral domains that are finite are also algebraic fields. For some reason, a group with a prime modulus is also a finite ring without zero divisors and therefore a finite integral domain and therefore also a field.

To be hard to attack, the modulus needs to be large enough and one needs to be sure that the modulus is actually prime and not just pseudo prime.