STM32 and SDCard: space management
No, when you write less than 512 bytes, your FatFS (or almost any other) library will:


Read the block into RAM
Append your data to the existing one.
Erase the block on SD card.
Write the whole block from RAM.


And I'm obviously not mentioning the library reading any file descriptors or anything which will most likely also happen if you open the file every time you want to write to it as opposed to writing to one that's just opened once at the beginning.

As to the second question: It depends. 
If you configure the DMA to use just a 512 byte buffer 2 things can happen:


If the DMA is configured to use the buffer as a circular buffer, in your example it will write the 2 bytes at the end of the buffer and overwrite the first 8 bytes at the beginning of the buffer with the remaining bytes.
If you're not using it as a circular buffer, assuming that you configured it to trigger a "Transfer Complete" interrupt, it will be triggered right after the last (512th) byte is stored in the buffer. Now depending on the transmission speed of your source of data (whatever it is you've not mentioned) you have a limited time - you must reconfigure the DMA to use another buffer or if there's enough time, process your data before the next byte arrives.
If you don't handle that Transfer Complete interrupt in short-enough time and new data arrives, all that new data will be lost and the DMA will trigger an Overrun interrupt (if enabled).