You may use all sorts of pumpkins and squashes (a Cucurbita moschata or Cucurbita pepo may be called either, depending on variety) to make a pie. The Halloween types may not be the best choice: they tend to be stringy, not very sweet, and sometimes over treated with pesticides. Instead look for the small varieties (around 5 lbs.) called Sugar, Long Pie, or Trickster.
When making the pumpkin purée for a pie from scratch it is important to deal with the water content of the pumpkin. After you bake, cool, and purée the pumpkin, you may need to use a cheesecloth to wring the purée.
It actually depends on the quality of the shortening you're able to get. You might think they're all the same, but you'd be wrong.
When I was taught how to make pies, we used shortening, and the crusts were perfect. However, I was told that for home baking, the shortening you buy in supermarkets (Crisco, normally) just isn't going to cut it and to use Tenderflake (lard) instead. I actually verified this once and found out that he was right - using the exact same technique, the supermarket shortening just didn't turn out the way the "industrial" shortening did. The crust is always too mealy and dry and tastes "off" somehow. Unless a lot has changed in the past 5 years, lard is actually much closer to the good shortening that's being used in bakeries.
Butter would, obviously, impart a much richer flavour than shortening, but I wouldn't use just butter in a fruit pie. You won't get anywhere near the flakiness of shortening or lard. Half-and-half is a decent compromise, but the result is neither as flavourful nor as flaky as lard.
So, generally, I would stick to lard. You could take plor's suggestion and mix it with some butter, but I've found that the flavour and texture is very good with just lard; if you do decide to mix, be careful not to overdo it, otherwise you'll lose all the wonderful flakiness that the lard imparts (I'd recommend 75% lard).
Don't mix lard with shortening. That can only take away from all aspects of the quality, and unless you're worried about nutrition (in which case, why are you using lard at all, or eating pie for that matter?) then there's absolutely no reason to "taint" the lard this way.
P.S. Don't forget to add some sugar. A little goes a long way in pie crusts.
Best Answer
Use the tapioca starch, or, if you cannot find that, another tuber starch. It is more waxy than cornstarch, and therefore better suited for a glaze. It will give you a glossy, transparent appearance.
Cornstarch is a mealy starch, with high amylopectin content, and creates a soft, fluffy, somewhat cloudy mass, not as glossy or as transparent as waxy starches. By the way, it shouldn't have an off-taste if you cook it through.
I guess that most recipes haunting the Internet use cornstarch because it is so much more common in a home kitchen, and the home bakers found the result good enough for their purposes and/or were not aware that tapioca will give them a better end product. As you describe that you want a really good glaze, go for the tapioca.