Just added some cinnamon sticks to apple cider that was in a better bottle for 3 weeks. The airlock started to bubble once every 20 seconds, does this mean I have re-started fermentation? I plan to leave it in there for another 2 weeks.
I concur with Fire.eagle that your fermentation may have not started again, but it could have. The only way to know for sure is for you to compare gravities before and after the suspicious times.
A possible reason it may have restarted fermenting. The yeast may have stalled prematurely because they were cold or stressed. When you added the cinnamon sticks (and maybe moved your bottle) you might have disturbed the yeast enough to reactivate them. This is even more probable if you have experienced a recent increase in temperature such as Spring.
A side note: There is some sugar in cinnamon sticks. It is less then 1g per stick Source, so you would have to add a lot of cinnamon sticks to have a noticeable fermentation. Those sugars may also not be fermentable.
Not necessarily. The addition of cinnamon sticks might have provided a nucleation site for the CO2 that was already dissolved in your cider from fermentation.
Really, the best way to answer this is to take a gravity reading and check to see if it looks like you still have sugars left in your cider that can be fermented.
Edit by request for more details: I'm not going to try to explain nucleation properly, as it's been years since I've had to deal with that scientifically. But, basically, in this case, it's acting as a site that is easier for CO2 bubbles to form on than it would be for them to form in the cider. So, you get some CO2 that wouldn't otherwise leave the solution forming and bubbling out. Think Diet Coke and Mentos. It's the same principle; the Mentos acts as a nucleation site for the CO2 in the coke which causes the gas to leave the coke in a much quicker and more entertaining fashion.
It shouldn't have any real effects on your cider one way or the other. If you had wanted to make a sparkling cider, you'd still need to prime it with sugar later in the bottles; the loss of CO2 here wouldn't hurt you.