I have used partitions for decades and swear by them. It will not make your PC run faster. I use it for separating windows from my documents. C: is for Windows, D: is for my documents E: is for my installed software. If Windows goes A over TIT as it can do, I can use an image to get it back.
Without knowing the setup you have, I would just format the partitions that say recovery (Quick format should be fine SEE NOTE), install windows in its own partition.
NOTE: You might have some valuable documents in the un-readable partition which could be recovered by some software but it is a difficult process and may not work.
If you want to delete them though then just go ahead. You will need to do a format afterwards. Go to Control Panel, Administrative tools, Computer Management, Disk management.