The standard load routine will not handle this.
If you need to load under $D000-$DFFF you will have to handle it yourself; you can still use kernal routines (such as OPEN, CHKIN and CHRIN) to do it but will have to buffer and transfer or set the bank configuration for each byte stored.
Loading under BASIC ($A000-$BFFF), $C000-$CFFF or KERNAL space ($E000-$...
No it doesn't adjust the memory configuration - which is why loading an 'executable' file (i.e. start address of $0800) bigger than 202 blocks (about 50 KB) using a standard kernal will crash a C64. Same for your 8 KB/$C000 example.
Yes, actually! Probably not very many early/launch titles, but some later ones did. According to Donkey Kong Country's internal header, it does indeed use FastROM. The same can be said about Tales of Phantasia.