It looks like it might be a member of the cordyline family? If so, they're native to NZ and although do ok in mild winters don't cope very well with the cold winters we can get. I can't offer any care advice unfortunately, but hopefully have helped with identifying it!
I think it's only certain varieties that grow taller, such as the cabbage tree. I know mum had a red one (but lost it the year before last over the winter) that she'd had for a few years and was about the same height as the one you have there, maybe a bit shorter. Hope its ok, they're lovely plants and that one looks really striking in the middle of your lawn.
I don't think it would do any harm. I have a dracaena houseplant which is also a palm-type plant and as the leaves die off I just pick them off and throw them out - they always tend to be near the bottom like yours, so it could be just older growth that's dying off. Might be worth giving it a feed of some kind once there's a bit of heat in the air. Let me know how you get on!