The art might be fine, might be worth doing a test stream just to see how the laptop performs while streaming and making art at the same time. If it's running Windows 8 and struggles with Legacy of The Duelist, I wouldn't even bother trying to stream while playing a game.
This laptop also has issues connecting to Wi-Fi, so I doubt I can stream on it in it's current set without possibly setting up a wired connection and upgrading the laptop's own inner components in some way.
When they say they'll stop supporting windows 8, it's likely referring to compatibility with future updates, so you should be able to just avoid updating them and it should work fine. However, if the program uses cloud features or often makes queries to a server (Like Adobe's Creative Cloud suite) that may not be the case. I recommend looking at what the end of support messages for each program says.
Unlike desktop computers, almost every laptop is designed such that it is not upgradeable by the end user and seeing as yours is running Windows 8, that is almost certainly the case. You basically have no choice but to buy a new one.
So you're basically saying that I do, indeed need to acquire a new computer for the programs I use. The main program is Steam, Steam will stop running on Windows 8 in 2024. I won't be able to use RPG Maker without use of Steam. No Steam means no RPG Maker, no RPG Maker means no work on mod updates and no more work on Cycle. This is why the Hitomi update is being worked on, this may legitimately be the last update I can possibly release. And now that Hitomi is done, I need to work on Coach Mary, because if this is the last update, it'll be one that finishes as much of the story as it can!
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As I've said before, I am very tech illiterate so I have absolutely no idea on what would be a good computer to get. I have no idea what I should be looking for from the listings on this page. Can I just get the name of a specific computer model that would be useful for me?