Can I install the GRUB bootloader to my primary hard disk's MBR and boot into Linux from an external hard drive when connected, and when it is not to automatically boot into Windows?

let me get this straight. you want to boot from an external drive with grub. well I would install Linux on the external drive and put the grub there then ask bios to boot from your external drive. the problem with grub booting to external drive is it needs to load drivers just to see the drive. I believe grub2 has been working on that. some systems have a choice at start up when you press F10. then it asks you what hard drive or device to boot from.
some systems you must go into bios and set it up to boot that device.
Then you have isolinux to boot from also that is another way.

what grub does is run on the mbr it is a small program that reads bios. this is how it finds the drives. since there are so many different drives today it will need the USB interface driver to load then load a driver to make the device work. this is done at the kernel level. not at the grub level.
so what I did was make sure my bios will boot from USB and then installed to the external drive. then boot from external drive.
so why can bios boot the drive? Because it will be seen to bios as a DOS {disk operating system} then bios will search for the first sector on the drive. MBR and that is where grub is.
you will find using a live cd to install with will make things easier because the drive will be loaded as scsi device sd device
good luck

Thank you, Drakeo.

When I installed Ubuntu to the drive, it would not boot unless I restarted my computer. When I shut it down and cold booted the computer, it would not load grub. But if I restarted the computer (Linux, Windows, BIOS, whatever), it would work.

Long story short, I do not want to have to annoyingly restart the computer anytime I want to boot Linux. I want to find a way to fix this, and I am thinking if i can configure GRUB to be on the primary MBR, BIOS will find Linux.

Please help figure this out. Nobody is helping me ):

If you must have linux on a external drive then put your grub there on the external grub. and now boot from the drive. your grub should now be able to point to the windows. Here is the problem I could care less about windows I have not used it in 14 years. except at hotels.
the problem is your usb drive will be seen as sda so the grub must point to windows drive partition and that would be sdb now.