Is it unhealthy if you cut whole-grains out of your diet and just eat meat, eggs, dairy, fruits and veggies (for quick fat loss?

Cut out fruits... if you have to have a fruit once in a while, berries are ok, but i wouldn't eat ,more than 1/2 cup once a week.  Dairy is good - full fat dairy and in strict moderation. No milk.  

Cook with healthy fats (no refined oils or crisco).  Good fats are lard, butter, olive oil, coconut oil, avocado oil.  Good food sources of fats are walnut, avocado, fatty meat and fish.  

You might be interested in the ketogenic way of eating.  Loads of information online.

It's unsustainable.  Carbohydrates are the primary source of energy for the human body.  It's hard to get them without starches, such as grains.

No, but there's no need to cut out whole grains. It's the simple carbs you need to limit (e.g., things made with white flour, white sugar, white rice).

.

Fat loss shouldn't occur at more than 3 lbs/wk, although 1 - 2 lbs is more reasonable. Any fat loss faster is almost always due to an unhealthy method, and thus not likely to stay off long term. Fat doesn't appear overnight, so it's not realistic to expect to lose it overnight. 

No,  but it will likely cause both diarrhea and constipation AND whole grains are not and can not make you fat (unless eaten to extremes).

The fiber fills you up before the sugars get to be enough to cause liver dysfunction. 

Whole grains are okay,  but not required. 

What IS required is insoluble fiber to digest properly and fruits typically don't have enough and vegetables are some do, some don't,  but legumes are then required  to achieve fiber goals.