You need about 150 videos.
With this particular question though, I would say it's better to learn as early as you can. You won't have much data to work with having only 5 videos, so I wouldn't "worry" about analytics, but I would most definitely LEARN about analytics as early as you can.
There is a LOT involved with analytics, and I feel if you can begin wrapping your head around the basic concepts of analytics, and where everything is, and how to read the data at a basic level in the early stages, it can help you slowly understand your content and your audience viewing behaviour over time.
You won't have much data to work with in the begging, thus not worth WORRYING over, but it's definitely beneficial to learn things like impressions, click through rate, video completion percentages. That way, once you have more videos and more data to work with, the learning curve is far less overwhelming.
I think deep diving into analytics at 150 videos could be very overwhelming if you've never looked into analytics. I've been constantly learning how to read and understand my own analytics for years now, and doing it early on can help you get more comfortable with the whole process.
I'm about 2 weeks in, I have 5 videos, I've followed most of the best practices, I think the content is good. I think I'm worrying about the analytics too much. Is there an amount of time before YouTube just figures out what my channel is about and starts showing it more in search results?
Also, if I updated tags and description and my impressions drop like a rock, am I doomed or is YouTube just trying to figure it out again? Thanks!
Don't stress yourself over the data, just learn how to read the data. 5 videos in is still very early. Damon is right that making more videos will naturally get things going naturally. There's no set amount of time, there's no specific number of videos, and if you update tags and descriptions, and they DO drop like a rock, simply change them back. Whenever I change something in my videos, I let the new data sit for about a month before I decide weather it had a positive, negative or neutral impact.
Not every change will have a massive impact, but learning how certain actions, effect certain analytics, such as better keywords, boosting your search impressions, but also remember that search impressions don't guarantee a click, they just get your thumbnail in front of more eyes, it's weather the topic of the video it'self is interesting enough to click, and if the thumbnail and title convey the value of your video which will ultimately decide how many impressions end up getting a click.
It's something interesting to learn over time, and again, I'd start learning early on, so that it's less daunting later on. But also, at the end of the day nothing beats good content. And the best SEO in the world can't save a bad video (and your audience retention graphs in analytics will help you see weather the beginning of your videos are captivating enough, or if you need to recraft your next video slightly differently)