Well if you are using whole hops, then you can strain them out as long as whatever you use has been sanitized. With pellets, some will get caught up in the protein material (aka break) and the rest will settle out in the fermentor. You just avoid siphoning (racking) the sludge (trub) on the bottom when you transfer to either bottles or a carboy.
As for leaving them in, it is not a big deal really, iirc some people claim it makes a difference, but when I used pellet hops they stayed in and the end product was fine. The hop pellets will settle out by the time you are ready to transfer. I would advise against a bag because you want the hops to move around, getting exposure to the boiling wort and thus helping to utilize the full potential of Alpha Acids. I am not sure how well a bag would work with pellets, but it would be a mess to clean the bag.
My method now is whole hops, and the cooled wort is strained through a sanitized aluminum tomato strainer. I have very little trub as most of the break gets caught in the hops. I also have very little waste using this method, I just let the last bit drain in the strainer.