I want to know if it is typically always possible to sell at a moments notice? Will people still be buying when the price of a currency has just risen? Is that even how it works or must be done?
You can buy or sell at any time of the day. It's nuts. Go to sleep knowing Bitcoin is at one price and wake up and it's up 20% (or down 20%). It's truly a global market.
What will happen that you need to watch out for is (1) how long it takes for your trade to settle (when selling to fiat) and (2) is the exchange confirming a sale price that coincides with when you clicked "Sell!" Settlement should take a few days, that's normal in the investment world. Some exchanges may take a week. If you're not getting the same price you saw when you clicked "sell" you've got a problem, the exchange's platform is weak under high volume or they're playing games with you.
E.g., people complain about Coinbase for this exact reason...but it's hard to know if the few that complain represents the anomalies or a trend.
Can one exchange between a less common cryptocurrency and BTC, ETH or another major coin at any point?
Great question, yes but not all exchanges deal in all currencies. Here's a website to help you decipher which exchanges trade in which currencies:
https://www.cryptocompare.com/wallets/#/overview