I heard you shouldn't take fish oil with other vitamins, because it flushes the others out of your body. Is this true?

No. Actually, some vitamins are absorbed better when taken with fat. These include a, e, d and k. Fish oil (good fat) has been shown to be helpful with blood pressure and cholesterol, have anti-inflamatory properties, and might be useful in brain health and function. I agree that the very best way to get most vitamins is through your diet. The exception is vitamin d, which is hard to get from food.
No, not exactly... The most current research continues to show only limited benefit from nutritional supplements. The greatest benefit is to those who are already suffering from vitamin/mineral deficiencies due to poor diet. It is, and always has been, the concensus recommendation to obtain proper nutrition through diet. A proper diet will eliminate the need for supplements.
Unlikely. Fish oil is lipophilic, that is, it joins with other fats or oils when it disperses and forms a solution. Oils are absorbed in that form. The fat soluble vitamins include vitamins a, d, e, and k. The rest of the vitamins and minerals are water soluble and don't mix with it, anyway. That whole oil+water thing...