I think that Regina loves Henry as much as she is capable of love. She has already proved herself to be a far superior parent than her own mother (just by not using freaky magic on Henry — so far as we have seen). It seems to me that I have seen what looks like genuine hurt in her eyes when Henry clearly favors Emma over her. And honestly, why would Henry NOT favor Emma? She is hip and cool, and has never told him to eat his vegetables or clean his room, has never taken him to the doctor to get a shot or made him release the snake he caught and decided to keep as a pet. Emma is a buddy while Regina has done all of the nitty-gritty parenting stuff for the past 10 years.
As rumplegoldfan pointed out, in the real world, where we all didn’t know that the mother is an evil queen who has cursed the entire town out of spite and vengeance, the situation would be viewed much differently. Archie’s threat that Regina was going to somehow be in a custody battle holds about as much legal water as Mr. Gold magically making his assault charges disappear. Emma has zero rights to Henry. If it was determined that Regina was an unfit mother (which being cold and manipulative doesn’t even come close to sufficient), Henry would go into foster care and it is doubtful that Emma, with no permanent residence and little job history, would be very unlikely to be considered as a placement for him.
Legal realities aside, it will be interesting to learn the why and how of Regina getting Henry in the first place. My guess is that she was looking for someone to fill that void in her heart that Malificent predicted. And as we discussed in another thread, I think Regina will only be able to truly love Henry when she lets go of her hatred of Snow and learns to forgive. I am really hoping for some redemption there and not just Henry and Emma riding off into the sunset together. I would love to see them add some complexity (layers, if you will) to the Regina/Henry/Emma dynamic by having Henry realize that he does have some feeling for Regina and that, in her own way, she does love him. And also for Emma to really blow it with Henry at some point because she sees Henry more in a buddy/big sister way than a responsible adult/parent way.
(Okay, in the interest of full disclosure I should say that I am an adoptive parent and the whole adoptive-mother-as-evil-witch and biological-mother-as-savior narrative is not my favorite!! 😆 )