A:AnswerI would recommend something like the GTX 650 as it will run pretty much any current gen title albeit not at ultimate settings. The price is fairly within your budget at an avg of $120.
A:AnswerNo! It's honestly about the same. You should go with something a bit better if you expect to do some casual gaming. A R7 240 has about twice the performance and can run most modern games at low settings. And it has DX12 support.
A:AnswerThat may only be a limitation of your monitor. some older monitors will only run at 1400x1050 or 1600x900. Id recommend looking up your monitor's specs or getting an updated monitor first. You can always return the monitor if it doesn't fix the 1080 resolution issue.
A:AnswerWithout the expense of an external card adapter (https://www.youtube.com/watch?v=bP_8EYQ-2RA) and the know how to install it you'd be best served saving up to invest in a newer laptop with better specs. In short, no this card will not go inside of your laptop, as it is intended for desktop use only.
