As someone in the game design industry I can tell you it's very different than what you see in videos and such. The problem I see happening a lot is that too many people think that game design is gonna be all fun and games, when it honestly isn't. Game design is a LOT of work. Often so much so that you spend more time working on them and very little time playing them. That's not to say it can't be fun, you just have to enjoy what it is you do; creating.
To answer some of your questions,
1. No, it's not required. It helps to know enough around computers that you can navigate and find files easily but that's really it.
2. and 3. See below.
4. Yes I would. If I discovered something that held my interest more than GD, I would pursue it. I wouldn't ever stop GD completely but if, say, I decided I wanted to play music more than I would start learning that.
now, for questions 2 and 3, I'll answer them both together.
To test your skills, you have to play around a bit. If you want to be a level designer, I suggest playing around in UDK. You can download the engine for free but you couldn't publish anything until you bought a license; a great way to learn the engine without a money commitment.
Scripting is essentially handled by the level designers anyways because it's very light coding, and in the cases of Engines like UDK and CryEngine, it's handled almost completely by flowcharting style (Kismet for UDK, Flowgraph for CE. Both come included with the software)
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Programming is a different beast entirely and requires a lot of self study. I taught myself almost everything I know about programming through google searches, and a great engine to try that out in is the Unity3D engine. This is another free to try engine. This is the pickiest of all the game design positions since it's so easy for things to break even when the logic makes sense but arguably one of the most desired due to how important the position is.
One of the more prominent positions is 3D modelling. Level designers, character designers, riggers(sets up the characters bone structures) and and sort of physical modelers will need to be very well versed in these softwares. Industry standards are Autodesk 3DS Max, Autodesk Maya, and ZBrush. There's a lot of other programs used to help but these are the ones you'll find used the most. Unfortunately there's no free-to-try other than the 30 day trials for these. You could always try Blender as that's free, but I don't know that program so I couldn't help you there.
Things like voice acting aren't strictly game design and requires a whole different collection of software. I would suggest Pro Tools for recording software, though it's very expensive. Hardware I'm afraid you'd have to ask someone else.
UDK: https://www.unrealengine.com/news/febru ... r-download
It's also highly suggested you verse yourself in photoshop as it get's a lot of use as well.
Beyond this, the most I can say is start researching. Play around where you can and see what it is that interests you the most, and just start researching and studying. Those are the biggest thing's you'll end up doing in game design. I've been doing this for years and I STILL find myself studying more than doing in most cases.
The great thing about this field is that while it's difficult, it's incredibly well documented and there's tutorials for almost everything. Just gotta look.