Good questions!
There's some old sayings on Wall street - that can, and do, make "some" sense... as in "nobody ever went broke taking a profit".... True enough -- but -- Always seems to be a big butt right?
So your question is multi faceted and really complicated - but also somewhat easy --
If a company reduces or suspends paying a dividend - YOU SELL IT! FAST! That is the kiss of death because that means they don't have the profits to continue to share with YOU the guy that owns the company (via stock ownership).
When a stock is going down - you need to understand WHY it's going down - a sales slip? A shrinking profit margin? Accounting fraud? There's so many things that can cause a stock to slip -- or is it the "SECTOR" the stock is in and all in that sector are going down? Or is it just that we're in a bear market and EVERYTHING is down? That's why that is a hard question to answer because there are so many reasons. Having said all of that --- there's a lot of money to be made buying the "best of breed" stocks in a bear market.... and that's when I personally do my best "shopping" -- because it's like every stock in the universe just went "on sale". I love to buy when stuff is on sale!
Now -- the other question was "what if a stock went up 100%" --- Buddy! Your mouth to gods ear! 'Cause that is what everyone is looking for!

But -- there's that butt again - you always need to "rebalance" the portfolio... and if the stock has gained so much that you're out of balance - you want to sell some of it and buy something else (diversify!).... Yeah -- it can keep going -- and you'll kick yourself for not having held every dime when it doubles again.... BUT that's where the old saying "pigs get fat and hogs get slaughtered" comes from. Better to be a profitable pig. Think about it this way as well - if you had a double - and sell half - the balance is "house money". That is a very nice place to be!
NOW -- all of this "depends" -- depends on how much money you have - how old you are - how familiar you are with why the company is going gangbusters etc. I've had stocks that have been at .56 - gone to $28 - and gone back (faster) to $1.70 - and then gone back to $35.... Those are RARER than a ZL1 -- and not for your average Investor 102 class. So taking some of the profit off the table is the "prudent" course... even if you live to regret it. Scroll back and read my post about the 20 million dollar yacht (boat) I owned. But I've never looked back and have been happy taking the gains and using them to live another day.
Wanna look at a real life - recent - scenario? Look at the chart of NETFLIX (NFLX), there's a stomach churner! And that all came about because the CEO shot his mouth off... So to speak. One day you've got a 3Xr and in a manor of a few days and weeks - you're looking at a loss! But if you'd bought at 100 and sold some at 200 - and sold some more at 300 - you'd be "okay" holding here at 65 (I'd have bailed completely).