Dual boot installation Windows 10 / Linux Mint

The safest approach to installing Linux in a way that won't disrupt Windows is to use virtualization -- VirtualBox, VMWare, etc. Such tools let you run Linux within Windows (or vice-versa), so you needn't repartition your hard disk or mess with the delicate boot process. You'll also be able to run both OSes simultaneously, so you can do something in Linux at the same time you're doing something else in Windows. The drawback to virtualization is that you'll sacrifice some performance; the virtualized OS won't get full access to the hardware and so will perform more slowly than it would if it were installed natively.

If you decide that virtualization is not for you, it looks to me as if your Windows is installed in BIOS/CSM/legacy mode. I say this because there doesn't seem to be an EFI System Partition (ESP) in the partitions you've shown, and an ESP is more-or-less required for booting an EFI-based computer. To be 100% sure, though, see the instructions on this page of mine for identifying your Windows boot mode.

You say that you want to install Linux in /dev/sda1. Be aware that, aside from virtualization and one or two other exotic installation methods, Linux completely takes over any partition(s) you give it. Thus, if you install Linux to /dev/sda1, any data on that partition will be lost. If that's fine, you can proceed; but if not, you should re-think your plans or move your data elsewhere. If you want the three partitions you mentioned, your /dev/sda1 will be replaced with three or four new ones, depending on how you partition the disk. (Personally, I'd replace /dev/sda1 with an extended partition and then create root (/), /home, and swap as logical partitions -- /dev/sda5, /dev/sda6, and /dev/sda7 in Linux's nomenclature. There are other ways to do this, though.)

As to the nitty-gritty details of installation, there are numerous tutorials on this subject. Try Googling "install Linux Mint," read a few pages (or watch videos, for video instructions), and if you have any questions, ask the authors or post your specific questions here or on the Linux Mint forums.