How to seal gym floor mats along the edges to protect your floors with knobbles facing downward?
Just going to throw the relevant suggestions into an answer:
Sheet plastic under the whole floor I think is the most thorough and cheap option.  Well placed double sided tape will prevent a slipping hazard.
I second the recommendation of jwh20 in throwing down a layer of plywood or similar to reduce the risk of drop damage.  The crushing destruction of the wood fibre will absorb an immense amount more energy than the rubber mat. It is possible to inset a tile level with an existing floor, but it is a huge pain and it won't be many tiles that make you wish you'd laid plywood.
You can mostly lay them button side up as FreeMan suggests to make them easier to tape together, and I'd note if you apply it warm to clean surfaces, many types of tape are flexible enough that button side down isn't a huge issue.  Wider tape will work better as well.  However, taping long seams tightly on the underside of something is a huge pain.