Q: How to make partition in Seagate backup plus in Macbook Air?

Unless you need to retain the Windows files you have put on the drive and copied to the computer, then I suggest you simply re-prep the drive for use on the Mac. Because the drive was formatted on a Windows system you cannot partition it via the Mac. But you can re-prep the drive using GUID partition scheme for two partitions and format one of them for MS-DOS or ExFAT to use on Windows. But as I said earlier if you don't need to use it on Windows and wish to use if for backup on the Mac, then prep it exclusively for the Mac. Using Time Machine for backup requires a volume that is at least 2 or 3 times the capacity of the computer's HDD.

Drive Partition and Format

1. Open Disk Utility in the Utilities folder.
2. After Disk Utility loads select the drive (out-dented entry with the mfgr.'s ID and size) from the side list.
3. Click on the Partition or Erase tab in the Disk Utility toolbar depending upon which is activated. If both are activated, then choose Erase.
4. Use the adjustable Pie chart to set the number of partitions if you want more than one. Otherwise leave at the default of one.
5. In the drop down panel set the partition scheme to GUID. Set the Format type to Mac OS Extended (Journaled.)
6. Click on the Apply button and wait for the Done button to activate.
7. Quit Disk Utility.

[Permission to use part or all of the above has been granted by Kappy, exclusively, to theratter.]

No, it is not possible on the Mac. I don't believe you can create partitions on an NTFS formatted drive even with the Paragon plug-in. Furthermore, you would have to be able to create a Mac partition for use with Time Machine. I don't think that is possible unless you prepare the drive for Mac OS X. My suggestion is to get a second drive to use for your added storage needs (not used for backup.) It's a bad idea to use the same drive for basic storage and for backup. If the drive becomes corrupted or fails, then you lose both the storage files and the backup.

Yes, except for the part about creating a Boot Camp partition. That can only be made on the internal SSD meaning you would need to allocate a part of the 128 GB SSD to a Windows volume thus imposing a fairly tight limit on the space for both operating systems.

My personal solution has been to use a NAND-based PCIe SSD in an external enclosure that connects via USB 3.0 to the computer. For example:

Samsung 850 EVO 500 GB M.2 3.5-Inch SSD (MZ-N5E500BW)

And, an enclosure for it:

Oyen Digital M2S-U3-BK M.2 SATA to USB 3.0 External Aluminum Solid State Drive SSD

You can buy the SSD card in sizes from 128 GB to 1 TB.  The 500 GB SSD and case together cost $198 USD at Amazon.

If you are asking if you can simply copy your files again from the Windows PC to the Mac-formatted Seagate disk, then the basic answer is, No. Windows can neither read nor write Mac-formatted disks without third-party software such as MacDrive. It does for Windows what the Paragon and Tuxera plug-ins do for the Mac so it can read/write NTFS-formatted disks.

But there may be another way to do the transfer by using the Windows version of the Mac utility, Migration Assistant. See About Windows Migration Assistant. I am unfamiliar with the Windows version, but the FAQ should explain the basics for connecting the two computers together and how to use Migration Assistant (both versions) to effect the transfers. If you find that you can do that, then you would be free to re-format the Seagate disk and not be concerned about losing your PC data. Here's a related article  to help: Switch Basics- Migrate your Windows files to your Mac.

It may be more a matter of needing to have two machines each of which has such a slot, and, how much data are being stored. I think 256 GBs is the largest SD and is pretty expensive. But there is no reason you can't use it for storing your files. I believe your MBA can also be booted from a system installed on an SD card.

As for using one in lieu of the SSD you will have to consider speed. They are pretty slow. OK for data storage, but kind of slow for use as anything more than an emergency system. USB 3.0 flash drive would be a better choice. I think you will find them cheaper and faster.