Adding ssd...how do i only transfer OS to ssd from existing HDD?

We do not know which Mac OS version you are running since your profile shows you are running iOS (which does not run on computers - only on iphones, ipads, etc). So it is difficult to help.

If you intend to make the SSD your bootup volume, then why not boot into recovery and simply download a fresh OS - and, I would do that before you install it. I'd put it in a temporary external enclosure, plug it in and download the OS. I would then install it and make sure that it is chosen in System Preferences as your startup disk. If you leave the other drive in, you will need to use the Option key to choose which one to boot into. (I also do not know how to delete or transfer just an OS and since some apps are part of the system, I don't know how well they'd run or not without it).

You can clone the entire HDD to the SSD. I don't know of any way to transfer just the OS, other than using a cloning tool to omit major folders from the clone (but still maybe transferring a bunch of settings and folders that are not strictly part of the OS). You could do that back with the real "MacOS" (vs. what they now call MacOS) OS9 and earlier but OSX+ has all kinds of hidden stuff. You can, of course, do a fresh OSX installation.

How to reinstall OS X on your Mac - https://support.apple.com/HT204904

The easiest method is to boot into the Recovery Partition by holding down Command + R on startup. The open Disk Utility and format the SSD. After it has been formatted correctly, the exit Disk Utility and use Install Mac OS and point it to the SSD. After you have done that re-boot normally and open System Preferences - Startup Disk and select the SSD and then reboot. The down side is you will still have Mac OS installed on the internal HD.

Yes it should, however how I'd do it is once you have OS up and running on the new SSD. I'd ensure I have a VERY sound backup of everything on the internal HD. Once that was confirmed I'd do and erase on the internal HD to wipe it 100% clean and then use my back up to restore on the data files needed back to the internal HD. Then I'd use the SSD for just my OS and applications and use the HD for storage. That's what I did with mine.