Grammar/Punctuation/Spelling, etc.
The animal clinic looked more like a cottage than a place of business, and a bell suspended from the door, chimed when I pushed it aside.
The comma after door isn't needed, as far as I can see. It interrupts the flow of the sentence.
"Pleased to meet you," I hesitated and then took her hand, "Mrs. Whittaker?"
The section I hesitated and then took her hand is an action, not a dialogue tag, so the commas should be full stops.
The authorities would come, and when my baby was born take her from me.
Since when by baby was born is an introductory phrase, consider a comma prior to it and after it.
With one firm hand around my upper arm, she pulled me up, and headed toward the back door.
Since the same person performs both actions, the comma after up isn't needed.
Mrs. Whittaker sighed and opened the car door, . . .
This sentence needs its own paragraph so that the dialogue from Alexandria and the actions of Mrs. Whittaker aren't combined in the same paragraph.
When I woke the next morning, . . .
Since this is being told in past tense, and this sentence is about the past, it should be, When I had woke the next morning so that the reader understands it is what happened before the start of the story.
I endured the icy jelly squishing under the doctor's instrument, while she listened to the heart beat and watched the monitor.
The word while is a subordinating conjunction, so no comma is needed.
"A child possessing the halo effectā"Doctor Jennings, as if considering my age, . . .
Just need a space after the quote mark before Doctor.
What did he mean? It's the only way?
Consider placing this in one sentence. For example, What did he mean, It's the only way?
"It's one of the original homes, built in the 1800's.
Consider removing the comma here for a smoother flow.
Years agoā"Mrs. Whittaker . . .
First, there needs to be a space before Mrs. However, I think using an ellipses would work better here to show that her words fade away, rather than being cut off.
all the way to the back of the house, where a solid glass door overlooked a garden, illuminated by hundreds of miniscule lights, . . .
The comma before illuminated isn't needed because the garden that directly precedes it is what is illuminated. The comma ends up separating the noun from its verb.
I sucked in a breath and cradled my baby bump.
This sentence needs its own paragraph because the dialogue and other actions are all Mrs. Whittaker's.
Becky squeezed my hand and then released it. "Promise me you'll think it over. You might change your mind."
New Paragraph here so that Becky's dialogue and action isn't mixed in with Alexandria's.
"She asks where Auntie Alex is, every single Saturday.
No comma here.
before the graveled driveway appeared, where several cars were parked.
No comma before where here because it is acting as a subordinating conjunction to combine an independent clause and a dependent clause.
I stole into her office and Googled 'halo effect."
Just need to be consistent with which quote mark you are using here.
"We'll be closing shop now, Miss Alexandria's ready to have her baby."
These are both independent clauses, so the comma should be a full stop or a semicolon.
But, yes, my instincts say the child is an extraordinary being, gifted by the Halo Effect.
Just missing the end quote marks on this dialogue.
blessedly missed the first demonstration of the Heaven's abilities.
The word the is extra here.
but the wariness in her eye spoke otherwise.
The word eye should be plural.
Keep my mother away from this child.
Consider placing this in italics because it's a direct thought.
I'll bet my mom has plenty to say. What if the event hits the news?
These sentences are in present tense, but they should be in past. For example, I'll bet my mom had plenty to say. What if the event hit the news?