Great examples. Let me try my hand at translating your examples:
うちの近所にはお受験に熱心な母親がいる。
There are moms in my neighborhood who are really into preping their kids for the school entrance exams.
(instead of "really into" you could say "obsessed with" if you wanted to sound critical of the time they spend on this)
彼は来年大学受験なんだ。毎晩受験勉強大変だよね。
He'g got college entrance exams next year. Man, that's gotta suck to be studying every night (young male talking to a friend)
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アメリカで医大受験は「MCAT」という試験。
In America you the MCAT is the entrance exam for medical school.
That should be "he's got" by the way - sorry for the mispelling.
Get has a LOT of usages. In this case, it means "he will take":
I have a test this Friday, so I have to study all week.
I've got a test this Friday...
She's got a test at the end of the month that she has to study for.
Hope that helps (HTH)
When I really think about it, "have got" is kind of slang used in speech; I would tend to use "have" in writing.