"Why did Lucy see herself still beautiful in the mirror when she wasn't, but Dracula see himself as his age?"
I interpreted that to mean that vampires saw what they wanted to see. Dracula presented a handsome face to the world, but wanted to see his true face in the mirror, whilst Lucy wanted to see her own beauty.
"Why didn’t they go to Whitby?!"
The foundation was in Whitby so I guess that was the nod to the original.
"But - if she is a vampire... a vampire can regenerate and heal all wounds by feeding off humans and lying in a bed of soil from their native land. That's how he stops being old. That's how he gets better after being set on fire and later blown up in the second episode ... so why couldn't she just regenerate?"
I think that drinking blood regenerates them to the point at which they became a vampire. The point at which they turned into a vampire is the best face that they can present to the world. So, for Lucy, her partially cremated body was what she was stuck with.
I liked the modern setting and the story was certainly new and unpredictable. But I found it unlikely that he would suddenly decide to embrace death, and I thought that vampires couldn't commit suicide anyway (Jonathon couldn't with the stake). Mind you, the writers are already hinting at a series so maybe the unlikely ending is meant to be full of holes.