Thanks for all the information, people. I've been to my local computer shop and the guy there totally brought me up to speed with things. There were so many things I was unaware of. Let's go through this again.
1) The option of upgrading the i3-2100 - After asking for your advice, I've also talked to the shop clerk. He told me it would be unwise to sink any more money into my old rig because of the things you have told me. Old processors would still cost just as much, new CPUs have perks.
So I'm no longer going to consider this an option.
2)
New budget - new setup - After talking through a lot of things regarding hardware, I've changed my mind how I approach this. He suggested the following build (more or less). We still didn't talk about coolers. I don't think I will need a HDD in the case, since all of my media is stored on an external drive. The setup is closely following Deadpool's build, with minor changes (GTX-1060 as a better future investment, an SSD to speed up the machine's load times).
https://pcpartpicker.com/list/MXbM6X
3)
How much RAM - Seriously, even for my i3-2100 build I've bought 8 GB of RAM just to make sure I have enough. Should I go for 16 GB for a better future investment? Or as a more general question: how do you even decide how much RAM you're going to need?
4)
The monitor - how quickly have times changed. Because of my new graphics card, I'm going to need a new monitor that supports HDMI. A VGA-to-HDMI converter would roughly cost me 1/3 of the price of a new monitor, so I would rather get a new monitor rather than a converter.
I am still going to use my i3-2100 since it's still better than my laptop, and I will save up for the i5 build.
Thanks for the advice.