Not without busting out the tile around the drain. Sadly, this is one of the shortcomings of square floor drains -- you simply have to unscrew it to open it and the unscrewing process for a circular drain is much simpler and won't necessarily disturb the tile you've laid around it.
The new ones don't require screws. They snap on.
Yes they do come in brushed nickel.
Store SKU #1003308396
About 3" hole through the floor is needed
the barrel is chrome plated brass.
I would search homedepot.com
We do not have replacement strainers for this drain. 423222 is the oil rubbed bronze option.