I blended a whey protein shake for the first time and it came out very creamy and smooth. I didn't seem as though it would seperate if I let it sit there for awhile. My question is, could I blend a shake, then put it into a travel mug and drink throughout the day or is this a bad idea?

I don't see why not. I always make my whey protein shake at night even though I'm not going to drink it until the morning because I like it nice and cold. So I just leave it in the refrigerator over night. I have to shake it thoroughly though because the whey does settle to the bottom a little bit.

Ditto.
I got sick of my shaker cups blowing their lids off or otherwise failing and getting on my truck's leather. Now I dump the water from a bottle of Arrowhead into the shaker cup, make the drink (no blender), then pour it back into the bottle and cap it.

If it's on it's side, altitude will still force a leak, but overall it's worked well. It does settle out, pretty quickly, but with the clear bottle I can see if it's mixed well or not.

If you just put it in a travel mug that you can't shake up during the day, you're going to end up drinking mostly water/milk and be left with "mud" in the bottom.