I assume you're using the current m.2 slot. Is it a fast drive, or would it be better to just get a bigger, better drive and clone to the new drive? You only have PCIe 3.0 to work with, of course.
As far as performance and functionality, a PCIe to NVME adapter will work perfectly fine, and in that slot, will be the same as an m.2 connector on the mainboard. You can even get a card that would take multiple m.2 drives, because you have x16 lanes and each drive only needs x4 for full performance (though you have to check whether the slot does bifurcation which it probably doesn't, so you need a card that does if you want multiple drives). But of course, you no longer have the use of the x16 slot for a video card.
You would not want to use the x1 slot even if it was free, because you'd be cutting the drive bandwidth to one-quarter of maximum. Most adapter cards won't fit in an x1 slot anyway because they're made for x4 or higher.
Get an adapter card rated for PCIe4.0 or 5.0, so that you have the option to use it for the future, even though your board is only PCIe 3.0. You can get a 4.0 drive for basically the same price as a 3.0, so that you could move it to a new machine and gain that performance without buying a new drive.
SATA would also work fine if you're just using it for mass storage. It will take much longer to copy big files, but it's up to you whether that will result in too much extra time spent on tasks. An adapter for an m.2 drive isn't expensive, and the NVMe drives aren't much different from SATA drives either.
SSD type and size depends on what you're using it for and how much space you need. Are you trying to make it last a really long time or would cost be a bigger thing and you just want to get by for a while longer? Are you looking to store a LOT of stuff on it? Are they many large files, or many many small files? Is it going to have applications or games running from it, or just be data files?