Translation:My father is a doctor.
Sometimes yes, in this case, Mom and Dad (妈 mā 爸 bà) the second character of each, instead of having the same tone (first and fourth respectively), they have like a... Blank (empty, void, zero,...?) tone. I don't remember if this is a rule or something like that, but, if you try to pronounce it, you'll see that if the second character had the same tone as the first one, it would sound weird.
(I hope I explained it well)
Yes you can drop "的", just like @Lomochibi's comment, but it's not strange to use "的" , for example: 这是我的女朋友(This is my girlfriend). This expression is slightly formal than the expression without "的": "这是我女朋友" .
Besides, you must drop "的" when the structure before "的" with only one character, for example:
新衣服(new clothes)；好朋友(good friends)；旧自行车(old bike)；大房子(big house)