I have heard that there are DC generators where resistance can be regulated by using electro-magnetic field. Where permanent magnets are substituted with electro-magnets. Are this kind of generator exists?
Look up for "dynamo".
There are DC machines without magnets. The function of the magnets is, as you said, replaced by a field producing coil. They are know as "separately exited" machines, you could take a look in here: http://www.electrical4u.com/types-of-dc-motor-separately-excited-shunt-series-compound-dc-motor/