I can think of two possibilities:
1. Hardware accelerated custom crypto algorithms.
2. Specialized co-processor. You could implement an exotic processing architecture on the FPGA to off-load computations to, like a complex number processor with 52-bit words in a custom floating point format for a specific application.
Nope. Reconfiguration takes milliseconds. Also, Arria 10 supports partial reconfiguration which means that the FPGA can keep operating while some of the logic is reconfigured via bitstream.