Marlin has write functions, but they are never called in normal operation. You pretty much have to use custom commands or modify the firmware to write to the card.
As for SDHC, or SD. SDHC can work, but not all manufacturers implement the SPI protocol of SDHC cards correctly, causing problems with some brands. And even within the same brand there are differences. But the card that comes with the UM2 or UMO is very well tested, so that one should work fine.
If you want to know how messy SD card land is. Just look at this compatibility list from another project:
http://www.wiibrew.org/wiki/BootMii/SD_Card_Compatibility_List
I have a card that gave random problems with Marlin. Made a 1:1 dump, tested the whole SD code in simulation on my PC, no problems then. So I suspect the problem is more with the actual cards then the code.