I see you've already done something, but for future reference, adding more details to what Fishrrman has said: when in Disk Utility, in the upper left corner, click on "view" and when the menu opens, click on "show all devices." Then when you highlight the main drive, you will have the choices of naming the drive, formatting the drive to Mac OS Extended [Journaled] and setting the Scheme to GUID Partition Map. Then click "erase" and the drive will be appropriately reformatted. No need to be mucking around in some Windows program -- AAGGH!!!
I do not think it is a good idea to have Time Machine or any backup on the same external drive along with other storage. External drives can fail, too, you know..... A better solution would be to buy another external drive and use one solely for the purpose of backups and the other for the purpose of supplementing the computer's internal drive. For one thing, the general rule of thumb is to have the backup drive with at least twice the capacity of the internal drive. In making a partition of only 350 GB for Time Machine, that is likely to create problems in the future. Drives do fill up eventually over time, even with incremental backups.
I've never been a big fan of Time Machine, as it can be quirky, and, right, as you state, it does not create a bootable clone/backup. Also, both SuperDuper and CarbonCopyCloner are more flexible and offer more options in how you want to handle backups.
The way I manage backups, storage and so on is to have external drives devoted solely to backups and archiving, and other drives for what I term "supplementary" drives; the latter are external SSDs which hold all the stuff I want easily available but which does not need to be in my computer itself. I can plug in one of those external SSDs and retrieve or look at any file or image almost as quickly as I could if it were still on the machine's internal SSD. When I do monthly backups I first back up what's on the machine, and then I take off anything that accumulated during the month that I might not really need to keep on the computer, and stick that into the supplementary drives, which are incremental. I back up the supplementary drives as well, so that nothing gets lost. Yes, this system necessitates multiple external drives, both "spinner" ones and SSDs; fortunately prices on external drives are fairly reasonable for "spinner" drives and eventually will be coming down on external SSD drives as well.