Sorry fellas I got lost there, is there a way to get a recovery partition then for Windows back and operable? I would be more comfortable with Linux if I had a fully working Windows side, complete with recovery E: drive. It sounds like Wizard what your are talking about would delete that in favor of a new partition for the Linux?
Do you have another computer (or phone) so that you can get back on here if you get stuck and your laptop won't boot? If the computer you are working on is your only way online, then you should be extra careful about which steps you decide to try out. Working with partitions and operating systems can be very hazardous. This warning extends to everything I'm going to tell you below!
My guess to your quoted question is: no, I don't think you can get a usable Windows recovery into your Recovery (E: drive) partition. I have been proved wrong many times before (ask my wife!) so you need not be too discouraged by my guess. You can try to copy all the files and folders from your USB into the Recovery (E: drive) partition... but the only way to use them, and hopefully initiate a system recovery, is to boot on that partition. With UEFI, this may be possible, but I have doubts about it being able to navigate into other partitions to find those files. You might want to try this.
At this point, do you think you will need the HP_TOOLS and the SYSTEM partitions that remain on your hard drive? You will note they do not have drive letters... that helps to protect them from accidental damage or deleting, but they can be deleted. Your original RECOVERY partition may have been protected in that manner too... I have seen that many times. But if it was protected, you got around it, and when you formatted it, then Windows assigned a drive letter and made it usable to you. If you had made a Recovery USB or DVD set with your original Windows 7, it would likely have restored all of these things for you, but that doesn't matter now... it's just a reminder for your next new computer.
We can spend a lot of time trying different things, but let me just give my suggestion for what I think is the quickest and simplest resolution to get you dual-booting. Get a beverage... this turned out longer than I intended. Sorry.
1. Back up anything important you want to keep.
2. Boot on and use the USB recovery stick to do a fresh install of Windows. You have not tested it, so you don't even know if it works. I think it is good to know it is dependable, and to know that it is indeed a fresh install. Maybe you have something else... maybe it is a "image" of the system when you made it... that would be different, and it might have programs you've installed and keep your settings. You should know what this USB is and what it does... it is your future Windows recovery, the only thing you've got... you should label it and keep it safe. If the installation asks, I would tell it to use the entire hard drive, which should eliminate the HP_TOOLS and SYSTEM partitions, but it might not. Or you could use Windows Disk Management to delete them at this step, if you want them gone. The HP programs that would call on these partitions will probably be gone after the format anyway (unless it is an image).
3. After the full Windows install, I would use Disk Management, as Wizard suggested, to shrink the Windows partition as much as it will allow, or to the size that you want to keep it. Do not format the "unallocated space" that will be created after the shrink... but note how big it is in GB. If the HP_TOOLS and SYSTEM partitions are still there, I'd probably just leave them alone. They are small and not worth much effort, but you could again delete them at this step (preferably before the shrink).
4. Check UEFI settings. Disable Secure Boot and Fast Boot (hybrid boot). Check your boot order (UEFI before Legacy if you see that option, DVD or USB before hard drives). Every UEFI seems to play with wording a little, calling things by different names, so ask if you have any confusion. Linux Mint should work with UEFI, but not Secure Boot.
5. This is probably important: After checking UEFI, go back into Windows.
Check this link and follow the steps to turn off Fast Boot in the Power Settings of Windows 10. This makes sure that Windows does not try to put the computer into "hibernation" when you shut it down (which is a problem if you try to load Linux next, instead of Windows, because Windows is still actually running in a hibernated state).
6. Boot on your live Linux Mint media (DVD or USB) and run the installer. Things are a little tricky here and you have to pay attention. I'm not familiar with every option from the top of my head, but I can restore Windows to a computer and then Mint to give you an exact play-by-play if you struggle with this. There is an option to "Install Mint beside Windows" that might be a good choice if you didn't shrink the partition in Step #2. Instead, I would choose "Custom partitioning" or "Do something else" (I forgot how it describes this option... but you get the idea). This is where you can choose that "unallocated space" you created as the place to install Linux. You have to recognize it (why I said to note how big in GB that it was). Double click on that unallocated partition and you'll get partitioning options. People usually create a SWAP partition first that matches RAM size... so 8GB for you. Then you create another partition and tell it what type of file system (usually Ext4 is good), and you have to give it a "mount point." For a new user, "/" is a good mount point. Make sure the bootloader is set to install to /dev/sda on your computer (it probably says MBR, which is okay, but probably not quite accurate).
7. Answer a few more questions along the way. Don't forget the passwords you enter! I do not recommend encrypting anything unless you have a serious need (more chance for problems). Linux Mint can install in about 20-30 minutes if you go quickly, but I really suggest going slowly and reading carefully, so maybe 30-45 minutes or so. If all goes well, you will reboot when the install is finished and come to the GRUB bootloader screen where you can choose either Linux or Windows. Linux will be the default, but you have 10 seconds to choose Windows. If you don't get the GRUB bootloader, then you may have to hit some F-key to bring up your UEFI boot options (sometimes have to use this to boot a USB or DVD too).
Now... if you really want a quick resolution: Skip to Step #6, and choose "Use entire disk" at the partitioning step. That will erase everything on the hard drive, including HP_TOOLS and SYSTEM partitions. That will kiss Windows goodbye as well and your laptop would be Linux only. But if you trust your Recovery USB, you could use it later to restore Windows again in the future. It's just hard for me to trust something I haven't tested.
Whew... that was fun! But
@wizardfromoz has lots of knowledge and good ideas, so maybe he will give you better advice and options to choose from. And
@Steve, and others here as well. Looking forward to seeing how you decide to go, but I'm pretty sure you'll be running Linux in no time.
Cheers!