One of my sites (3 years old), has ranked on the first page of Google for a keyword for over 1 year. Last week, it droped to the 5th page. Have I got a penalty?
That is a weird amount to drop man. Do you use 100 results per page or 10 like most of us? Dropping to around 500th would sound a lot like a penalty. Dropping to just 50 would confuse me.
There are no obvious signs of a penalty. But another keyword which was ranked around 120 is now not ranked at all. All other keywords are fine.
How often do you update your site with fresh content? How often are you promoting it? Google is updating a lot more frequently now, so the minute you stop paying attention to your site, Google will also stop paying attention to it and will start to look at your competition's site.
If your site is showing up anywhare in the SERP's, there is no penaly. Allow Google to check for you: https://www.google.com/webmasters/tools/sitestatus
Ok, so now it's not ranking at all for the keyword. It must be a penalty, right? But the site is still indexed, just not showing up for this keyword. Only reason I can think of is that I got a lot of links recently with the same anchor text, but the anchor text wasn't this keyword...
Again, how come not sharing the site name? My guess is Google changed its algorithm or something and this affected your rank but I'm not sure because I dont know your site.
What does it mean when your page has been slowly moving up, sits on page 1 for a few days and then drops all of the way back to page 10?
Did you check your Google Webmaster Console? If it was in fact a penalty theoretically you would be able to see a notification on your panel. Despite not really knowing what happened, I suspect that you dropped your position as a result of the normal "dynamic" of the Google's index update process. The SERPs are not static, and sometimes a minor update can temporarily shuffle the results. Usually things tend to go back to normal in a very short time span.
Did you have your server go down? If it were down for a while Google would not have been able to find the site and would there for drop it down / out of the index. Adding a blog to the site and making some posts for a week or so should resolve the issue if it were due to this. Link back to the page that dropped from the blog with your keyword.
I submitted a reconsideration request using webmaster tools. They replied saying they've reviewed the site. Does that mean it has a penalty? Or is it just a generic messege?
Generic message, but if you didn't have an alert on your console you should be fine. This does not mean that the console displays all the penalties in place, but it should display the ones that can be re-amended.