1. I have a pair of PSB M4U1s that I really, really like, substantially more than my Grado SR325i headphones which are a bit more expensive. I suppose that would be my recommendation. I got that suggestion from the Wirecutter. They seem to know what they are on about.
2. A headphone amplifier would help some, but they aren't strictly necessary. If you've never listened to headphones with a headphone amp, you probably wouldn't know what you were missing. You can always hold off on that and get one later. Whether plugging something into a TV would help probably depends on the sound hardware of the TV. My guess is that the TV isn't really made with high-quality audio in mind. You might just introduce more interference into the signal that way. You could always try it and see how it sounds though.