BIOS is very old, and it surely needed to be updated. But UEFI has issues for Linux users. The issues are getting better, but it can still be very difficult to work with, or work around. Part of the problem is that there is no "standard" followed by OEM's. So every different brand computer you work with may behave differently. UEFI has been around for 10+ years, but it was finally required by Microsoft beginning on Windows 8 systems so that the "secure boot" feature could be utilized to protect the system from rootkit-type malware. At that time, however, the secure boot feature was required to have an option to disable... and another setting usually called "legacy mode" was included. For a long time, Linux required legacy mode enabled and secure boot disabled. Linux is getting better at dealing with, especially the major distributions, but there are still problems (at least I have problems with an HP laptop!). Now, in the Windows 10 era, the option of disabling secure boot is no longer required... and it will be a bigger hassle indeed if OEM's choose to eliminate that option. I've not heard of that happening yet, but it's still early in the Win 10 deployment. The future may demand that you shop around for "Linux friendly hardware" if this is important to you. OK, enough on that! Whew!
With 2GB of RAM, I think you'll do well to run many of the regular or light distributions, some I mentioned earlier. They typically include LibreOffice already and GCC should be in all of their repositories.
1. Chrome/Chromium/Firefox... I think these all will run AdBlockPlus. Firefox (or Sea Monkey) is often included with typical distros, and sometimes AdBlockPlus is already included too. The Opera browser also has ad blocking capability.
2. Office suites are often included, as I said. You can install OpenOffice if you prefer, but usually LibreOffice is the default that is included.
3. GCC should be okay for you. I'm not a programmer, so I can't be sure, but I'd guess that your RAM is okay. Maybe hard drive space could be an issue though.
4. I had to Google DoomRL... I'm not a gamer. But I would think you'll be okay here too. Their website says you'll need a few other packages, and also mp3 support, but this should be okay. Some of the regular distros will ask during installation if you want "extras" like mp3 support, so the easy thing is to allow it then (if you have an Internet connection). Assuming that the gaming is very important, if you need help from those folks, you will be far better off using a regular distro instead of Tiny Core.
C++ for Dummies... again, I'm not a programmer, so I can't give any good advice, just some general thoughts. I'm sure C++ is still used to some extent, and maybe a lot, but it isn't one of the buzzwords that you hear when people talk programming. Java, Perl, Ruby, Python... those are some of the languages I hear discussed more often. It may depend on where you want to go with programming... heck, my company still uses FORTRAN!
If you want to be a Linux kernel dev... I think you'll need the C language. You'll need to investigate this item further, but I'm sure some of the folks here can help.
OK, time for a beer! (Or, I should say... another beer!
)
Cheers!