I have an unused, unopened AMD Sempron 145 and want to know if paired with a good motherboard and graphics card would it be able to play games similar to minecraft, counter strike, garrysmod, terraria and other small games.
If you are up to the adventure, there is the possibility that you could unlock a second core on that cpu. Just do a Google on unlocking the second core on a Sempron 145. If it doesn't work or you aren't happy with the performance you could always buy another cpu and put it in that board. That board will take an 8-core FX so it's not like you are dead in the water if the Sempron doesn't meet your needs.
I would guess that the Sempron will handle the games you listed just fine.
And you should know that AM3 semprons are actually AM3 *regor* Athlons II x2 with a core locked. You can get a cheap mobo, unlock the 2nd core and get a nice improvement. Also an small overclock, lets say about 3.2ghz will let you play all those games pretty well.
I must add that everything i wrote can happen if you are extremely lucky to get a fully funtional core and stable unlock.
But if you are able to unlcok an adiontal core, there are chances that it comes bad and only gives you problems as stability or undesirable heat.
The best recomendation we all can tell you is to get a cheap motherboard and a very budget card. Spending $150 i guess is the very top you should go.
That motheboard looks kinda good you can use it, but for me i would go Asus or ASrock, also looks like that Gigabyte doesnt have Core unlock feature ! Well... just for your knowing, that a single core PC wont be for gaming, unless you play old games.
A dual core CPU is sufficient for playing games. Sure a quad core CPU can be better for games... if the game can make use of it. The vast majority of games do not use more than 2 cores. The exception is if you know that you are going to buy games that can in fact make use of more than 2 cores and/or if you do some CPU intensive tasks that can make use of at least 2 cores; like video encoding.
As for the sempron... if you cannot afford to buy a dual /quad core AMD CPU then at least the motherboard you have selected can at least use a Phenom II / FX / PileDriver (perhaps) in the future when you have more money for a better CPU.
The games you have listed do not need more than one core... not 100% sure about Minecraft..