Can I install OSX on my full data SSD without losing data files?

I would try first with that adapter you mentioned, if you already have it. Removing the SSD from your old mac and installing it inside the new machine makes more sense to me, since there will be less bottle neck from the USB to SSD conversion, not to mention connections that with time, will wear and tear, even the USB ports suffers.

To start the machine with an external drive you would need to do the following:

*Startup Manager*

When you use Startup Manager to select a startup disk, your Mac starts up from that disk once, then returns to using the disk selected in Startup Disk preferences.

1. Press and hold the Option (Alt) key, immediately after turning on or restarting your Mac.
2. Release the Option key when you see the Startup Manager window.
   

If your Mac is protected by a firmware password, you can release the key when you're asked to enter the password.

1. Select your startup disk, then click the arrow under its icon, or press Return. 
   
2. If you press and hold the Control key during this step, your selection is saved in Startup Disk preferences, so it persists until you change it.

Now, installing the old SSD inside the machine directly, will give you less trouble in the long run. It’s possible that you will need a cheap adapter, but that’s not always the case. You shouldn’t need to reinstall OSX on the old SSD since it’s already there. However, I would strongly recommend that you back up all the important stuff on a different drive and proceed with a fresh install, then pass your programs and documents, etc. to the old SSD on the new machine. Preferably the latest version, unless you have a preference to a particular version of OSX. Of couse, I’m not sure if installing an old SSD in this machine might be possible if there is not SATA type connector, which I believe it does, since it shipped with a “Fusion Drive”, a fancy way of calling it a Hybrid drive, Small NAND and Big mechanic disks. Basically, crap. So check out the specs of the machine you wish to buy and check your options. I recommend pure SSD drives. Kingston has reliable and fast SSD’s for fair prices now a days.