Depends on what material you will be staining and what colour as they are not all created equal so I feel you should elaborate some more.

Depends on the home style, and the fatness of your wallet. Stain grade is an upgrade. NICE stain grade, like a quartersawn oak, and not junk pine, is and even bigger expense.

Not a simple yes or no question. Home type, interior finishes and stile all play a huge role. As does material type and quality,

I love stained trim, when its done correctly and is a good quality. It is completely up to you, and your taste, what you decide to do. However, you only mentioned baseboard. What about doors, and windows, and their trim, etc. They are all trim, and baseboards, should be the same, as all the other trim.

I'd avoid staining trim if at all possible. If you want natural wood, choose a wood for your trim that is the right color to start with. You will avoid myriad issues that way.

If you do stain, or better, dye your trim, be sure to go through the entire finishing schedule including all sanding and topcoat steps on your samples prior to assessing color.

And buy quality wood. A good finish sstarts with good material.

I love stained window trim, just love it. Good-quality stained wood trim looks rich and luxurious to me. Of course, color-wise you've got to coordinate with the rest of what's going on in the house.

I do not like stained baseboards. I don't like contrasting baseboards at all, no matter the material or color. Baseboards are funtional, not something to be highlighted IMO. I always paint my baseboards the same color as the wall paint.

I disagree with above. Baseboards are trim, and always look better painted or stained the same color as the rest of the trim.