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I used Google Translate for these sentences but I'm not sure they are correct:
Mom 's white dog 妈妈的白狗
My mom's white dog 我妈妈的白狗
And it is possible if I say
Mom 's white dog 妈妈的白色的狗
My mom's white dog 我的妈妈的白色的狗
I wonder whether I should use 的 or I can omit 的 sometimes.
Could you also provide me the correct sentences?
All of your examples are correct. In my opinion, "Mom 's white dog 妈妈的白狗" is the most appropriate.
Whether the sentence needs to use "色的" depend on the context. For example:
"不管白猫黑猫，能抓到老鼠就是好猫" translate to English "Whatever white cat or black cat, it is a good cat only if the cat can catch a mouse."
"我刚才在街上看到一个红头发的人" translate to English "I saw a guy with red hair just now."
"你能递给我那本白色的书吗？" translate to English "Could you pass the white book to me?". It is a little strange in this sentence, if you don't use "色的".
In general, I suggested that you'd better add "色的", if you are not sure. Because that is never wrong.
This is more of an aesthetic / rhythmic issue than a grammatical one. Technically you can keep adding 的 to make a chain as long as you like, but it won't be "nice". It's like saying "that" repeatedly in English, e.g. "the man that ate a dog that ate a fly that watched a cat that liked to browse stack exchange." Grammatically valid but not natural.
The general rule with multiple 的s is to only keep the last one, but sometimes you do have to use multiple 的s. It's usually worth looking at rephrasing the sentence if possible to avoid this (in writing), but in speech I would say just don't worry about it. Even native speakers say things that don't sound perfect, especially if they pause for thought in the middle of an utterance, e.g. "妈妈的那个白色的狗" isn't nice Chinese but people could easily end up saying it.
Also, there is a slight difference between 白狗 and 白色的狗. 白狗 sounds like a particular established category of thing, 'white dog', whereas 白色的狗 is modifying an existing category of thing, 'dog that is white' (although you would translate it into 'white dog'). This may seem obvious but there is a subtle but important difference.
The omitting of 的 follows a loose rule of minimal reappearance in the context you gave. The first two are the correct expression in Chinese. The latter two, though with correct grammar, would never make to any Chinese conversation, oral or literal.
Also, another general rule here is that the closer the relationship, the bigger probability the omitting. Note that Mom has closer relationship to My than white. As white has same level of relation to mom and dog, both 的 are left their to imply possession. 我 thus become adverb for 妈妈. As a union, 我妈妈 acts as the adverb for 狗 and so does 白色. Changing the position of the two adverbs does not change the meaning which reflects the relationship.
My: 我、我的
mom: 媽、媽媽
's: 的
white: 白、白色、白的、白色的
dog: 狗、狗狗
Remember a common rule: "Don't use 2 more 的 in one sentence."
So following all are correct sentences:
Mom 's white dog
媽媽的白狗。
媽媽的白色狗。
My mom's white dog
我媽的白狗。
我媽的白色狗。
我媽媽的白狗。
我媽媽的白色狗。
A special word you can't choose in this sentence.
媽的
媽的 is a dirty word.(You can't say "媽的白狗")
看那只白色的狗is completely fine. But if you have another modifiers, 白狗 is better in prosody.看那只白狗is okay, too. If one can say 蓝天 and 白云, you should feel natural to say 白狗 and 白猫. As to books, it's always (colour)+皮书 in Chinese, so it should be 白皮书, 黄皮书, 红皮书, etc. 黄书 and 红书 would mean something different, they modify the content of the book, not the colour of the cover.