Can I use double-sided carpet tape to stick throw rugs to a carpet pad?
I think that you're on the wrong track. You can purchase actual carpet tiles specifically designed for that use case. The linked one is a hideous green with peel and stick backing, other brands snap together at the edges. I believe you'll find the cost per square foot roughly equivalent to the IKEA rugs. Installation will certainly be way easier.

I wouldn't trust underlayment that wasn't stapled down. If you are interested in the tile idea I would add the padding into your budget and buy nicer tiles. (You may have better results at a dedicated flooring store than the home center.)

If you are dead set against carpet tiles and stapling both you may want to look into rug underlays, which come in tacky (sticky, not tasteless) no-slip versions. Rug underlays are quite a bit tougher to rip than carpet underlay as well.

It isn't a great option for you because of the column, but you can order bound carpet in any dimension. (Bound carpet has the edges sealed so they won't unravel.) It's slightly more expensive than 14 Ikea rugs taped together, but it's a lot less of a trip hazard. 

UPDATE - You should definitely invest in a hammer tacker and staple the underlayment down since it's an unfinished floor. One staple every couple inches around the perimeter of each piece to prevent it from sliding or bunching up. Leave enough room that the rugs can be taped down directly to the floor on the outside perimeter.