Does i7 render better picture quality than dual core (Does Intel HD graphics matter which is used in both)?
No, it doesn't. They both render in the quality you tell them to, however, rendering is a really computationally-heavy task, so rendering with an i7 will be a lot faster than rendering with a low-end dual-core processor.
And no, the Internal Graphics Processor (Intel HD Graphics in this case) will not matter, since rendering uses only the CPU. However, some applications for rendering might use your IGP (Intel HD) or GPU (your discrete graphics card, if any is present) to render an image, which will lead to a completely different result. Most consumer-grade CPUs do better at rendering than regular GPUs, and a lot, a lot better than IGPs (both in quality [because of better computational algorithms] and in speed, however, this does not apply to this case). So you should keep this in mind, as it is varies from application to application. (credit to @CliffArmstrong for the suggestion)
Does processor degrade after a short amount of time because I use them
to render videos? (videos use 100% CPU for minutes)
No, processors do not degrade. They are manufactured so you don't have to change them regularly. Check this answer for more detailed information.
If the application which is currently rendering makes use of multithreading, then newer processors which also have a higher core count would be able to perform the same task a lot faster.
For example, let us say we have a newer 8-core i7 processor and one older regular dual-core processor and let us say that each core has 2 threads. That makes them a processor with 16 threads and a processor with 4 threads. Theoretically, if the application made use of all the cores and we specify the image to be of quality 1080p (Full HD), the i7 processor would theoretically render the image 4 times faster than the dual-core processor (if all cores work at the same frequency in both processors). However, the image quality would still be 1080p, so they will render the same quality image, but in different time.
And while processors are assigned such heavy tasks, they start producing a lot more heat, which is what can be dangerous. Proper cooling is a must-have when performing such tasks, as @Tetsujin mentioned in his answer, or else your CPU could begin to throttle itself down in order to reduce heat.