Now, cards overheat if they get too hot, obviously. Majority of the times the time your card will overheat is if it's overclocked wrong, faulty hardware, or using a less powerful card. Pretty much like playing Crysis 3 on a GTX 560. Don't take my word for that, I wouldn't know how it would preform.
Wrong. Using a weaker card to play/do the same exact thing is not going to be a cause for added heat. It all has to do with the chipset, the cooling solution, and what you're doing with the card. We all remember the GF110 Fermi's, don't we? The 580 had a hell of a reputation for running uncomfortably hot, that thing not only was insanely inefficient but it also generated IMMENSE heat. The GF114-based GTX560 on the other hand ran MUCH MUCH cooler. on ANY card, if you don't keep it within its designed thermal limits, it will overheat. You can usually help augment this by using a case with good thermal flow properties(especially across the PCI area), and by manually setting a more aggressive fan profile for the GPU
A GPU can have a HUGE number of different uses for a system.
First, let's break it down. There's NVidia and AMD. Both make top notch products, with NVidia usually leading the pack in their driver support.
Both will have marginal differences in their gaming performance, currently it seems that NVidia is producing the top-performing cards as far as games are concerned. Both companies provide cards from the low-cost/low-end all the way up to the highest-end dual-card/single-slot solutions.
That being said, perhaps you're not a gamer? Perhaps you do a lot of graphical processing, or subject your card to heavy compute loads? This is where AMD excels, and is TREMENDOUSLY more powerful than NVidia cards for single and double precision compute loads, especially when coded under OpenCL. If you want an NVidia that will compete here, you generally have to look at the Quadros and the Teslas ($$) A perfect example here was when the R9-series came out and the bitcoin boomed. Nobody was hardly buying NVidias because they simply couldn't even hold a candle to the AMD's for bitcoin mining(very intensive), especially in terms of cost to build a profitable mining rig. This drove the cost of those AMD cards WAY up, and sold them out nationwide.
Your GPU can be utilized to accelerate processing in things like photoshop, which is optimized for CUDA(NVidia) and OpenCL(open platform, which AMD dominates) as well as video encoding and so forth. Since a GPU can (believe it or not) process drastically larger amounts of information and can tremendously speed things up when processing intensive things, you'll find a lot of programs are utilizing your Graphics hardware in order to provide you with a better experience.
back to gaming, though, generally your GPU is going to be used for this. Here, it is simply a matter of preference. You'll probably have a better experience with NVidia due to their drivers, but you'll likely get better performance for the same cost by going with AMD.
In some gaming rigs, people go with multiple cards in an SLI/CrossFire configuration. Now, you wanna think "two cards=double performance" right? In theory, this is how it ought to be, but in reality.....this isn't the case. Adding a second card will increase performance but not in a linear fashion. for example, a single card is 100% performance...a second MIGHT put us at 170(often times more)%...a third MIGHT put us at 190-200%, and a fourth MIGHT put us at 210%. You just never know. 660's and 670's scaled GREAT, for example.
In particular cases, some games are poorly coded for SLI/XF or don't support it at all. In the cases of laptops, an 860M sli configuration (which should run about like a single 880M or better) will only perform like a single 860M. Same applies to desktops. You can have four 780Ti's, but if the game doesn't do well with SLI...then you're looking at a TON of wasted power, and a huge waste of potential.