Got a few questions I was hoping people might be able to help with:
1) Is it best to evolve Pokemon first and then power them up, or vice versa? Or does it make no difference?
It makes no difference to their CP whether you evolve now or later. However, movesets are permanent, and set at random on evolution. You don't want to spend lots of time leveling a Clefairy if it turns into a Clefable with a particularly rubbish moveset. So generally, you want to evolve first to find out if what you have is worth training further.
As a rule of thumb, though, you won't want to start evolving/powering anything except 12-candy Pokemon like Weedle, Caterpie, and Pidgey, until much higher levels, because simply leveling your trainer will get you much stronger Pokemon fresh from the wild without having to bother wasting candy and stardust. I'd wait until at least level 12 or so, but if you're really serious wait until level 20, which is sort of the main "gate" at the moment. Leveling past 20 slows down immensely, so it is much more viable to power stuff up.
2) When you have a duplicate unevolved Pokemon (that can evolve), is it best to always just transfer the one with the lowest CP? I was wondering if you get anything extra for transferring Pokemon with higher CP. I know if you have a fully evolved Pokemon/Pokemon that doesn't evolve, it can sometimes be worth keeping one with lower CP if it has better moves.
You don't get anything extra for transferring higher CP Pokemon. In general, just transfer the lowest CP 'mon you have unless a) it is fully evolved and has a better moveset, b) it was hatched from an egg (they have higher CP caps), c) it has something cool you like such as XL or XS size which have no real effect but just add some personal uniqueness, or d) sentimental value (starters4lyfe).
3) How do you guys determine which Pokemon to place in gyms, which to challenge gyms with, and how much stardust/candies to invest in each Pokemon? How do you decide how many Pokemon to split your resources between? Do you pump one up to be a powerhouse, or try to make an even team? If so, how many strong Pokemon do you tend to aim for?
Gyms fall super quickly. In my morning trip, I claim all 4 local gyms, get my rewards, and then accept I'll just have to do it again tomorrow; so I leave something behind that I reckon can last up to an hour or so. You can only really judge that by your local area.
You'll want enough Pokemon to have SE against all the major types, both attacking and defensive. That's only really applicable to high levels, though. Again, at lower levels the best power progression comes from leveling your trainer and not powering your Pokemon. Later on, you'll be wanting some of the top tier Pokemon like Alakazam, Dragonite, Snorlax, Lapras, and so on, but don't worry about that now.