Don’t worry about the burning software: you can use what you like. If you use quality discs, the result will be good anyway
I personally prefer to use DVD Shrink with ImgBurn, but it is only a matter of personal choice. My preferred method is to save an ISO file on HDD with DVD Shrink and then burn the ISO with ImgBurn.
When creating the ISO, DVD Shrink will automatically create the AUDIO_TS and VIDEO_TS folders.
I also use deep analysis with dvd shrink, does it make a difference?
Yes, it makes the difference, but the final result is highly subjective: some people thinks that results are worst, other that are better and other that there is no difference. Only you can answer to this based on your personal vision of compressed movies and on the screen you are using. With a very large television, for example, you can see some artifacts, but again only you can see personally if results are acceptable.
Also, will surfing the net while encoding and burning with dvd shrink effect the quality or playback of the dvd?
No, there will be no effect on resulting quality, but the encoding time could be longer because the CPU will be used not only for the DVD Shrink work but also for other activities. If this is acceptable for you there is no problem at all
On the other side, if surfing on the net while burning cause frequent buffer emptying, it is better to stop temporarily until the burn is complete, just to reduce the probability that the burn will have a lower quality. In fact, even if the underrun protection mechanisms avoid to “fry” a disc if the buffer is emptied during a burn, this will introduce on the disc a slight amount of errors in the point where the laser has been shut down temporarily. Moreover, if the buffer is emptied frequently, the burn will require more time to be completed.