How long should it take to do a 5 mile climb like this? Is a 5 mile straight climb likely in cycling single stage races? How about multi-day races? Comparatively, how does this rank against climbs that you guys would do? How long would it take you guys to do this? Also what gearing should I have for this? I get stuck in my granny really quickly. I run a compact 38/51 with 175MM cranks and my cluster is 11-22. Sometimes I pray that my 22 will turn into a 27 but it hasn't happened yet.
Then part two, I have a long descent after this. I get up to 40 MPH on the downhill. I top out my gears and I can't even spin downhill anymore. I get cold both literally (very literally) and cold in the muscles as in not warmed up. What do I do technically? Go slower but spin?
It takes me about 30 minutes to climb this bad boy. I don't stop but I'm a turtle. My average climbing up this thing says 10 MPH but I feel like I could get off and walk it faster.
Also, I'm riding a Leader TT frame. Would TT make a difference as in it's heavier and slower?
That climb is only a 1600' climb. That's probably a cat 3 or 4 climb in the Tour, if it's rated at all (the rating depends a lot on what else is going on that day climbing wise- if it is the only climb on the day it'd be a 3, if it's a 17,000' day then it wouldn't even be rated).
It's a 5.2% avg grade. Not super difficult. If you're doing it in 30 minutes then you are climbing at a rate of 3200 ft/hr. That's not bad for a non racer. Given the questions you are asking you probably haven't been riding for long, if so you should still have plenty of room for improvement. Top TdF pros can get close to 6000 ft/hr for the same time period (Contador did that in the Tour this year). The "non climbers" (sprinters) in the race do about 4500 ft/hr. I'm old and not a very good racer; I did 3650 for 40 minutes in a race recently. Good local cat 1s (good climbers) did 4500 in the same race. Ft/ht (or meters, whatever) is a good way to compare climbing rates across climbs of different difficulty. It's often called "VAM" as well, which is an italian acronym coined by the guy who invented it as a metric.
Do you really have a 51t big ring and a 22t large cog? Those are unusual sizes. If you have SRam there are some teeth missing to make shifting easier, but you still count the missing teerth for gearing purposes.
If your low gear isn't low enough, buy a different cassette. It's not hard to change although you'll need some tools. Don't worry about what gearing other people use. Some riders are happy with pedalling at a very low cadence, others go faster at a higher cadence and need lower gearing. And a lot of people on the internets lie about their gearing because they think that having low gears means you're a wimp. In reality it's how fast you get up the climbs that is important, not the gears you use to do it.
If you are getting cold on the descent, bring some more clothes and put them on at the top. I have a vest and arm warmers on just about every ride except when it is hot in the summer. I often take them off on the climbs and put them on for the descents. If you practice you can do it without stopping.
A lighter bike makes a difference but not that much on a climb that short. You can check with the calculator at http://www.analyticcycling.com/. It's more cheaper and effective to lose any extra body fat first. If you don't look like a prison camp survivor you could lose some more.
try a 12-25 cassette... it made a huge difference with me. i don't spin when I am descending faster than my highest gear... I get aero and hang on. if you are cold get some leg and arm warmers. unzip your jersey on the way up and pull up your arm warmers. when you reach the summit pull the arm warmers back down and zip up the jersey.
tt frames aren't for climbing they are designed to go straight really fast.. consider an all around road bike.