I'd be very very surprised if it is not. SDHC have the same connections as the earlier SD cards. There is at least one object that lets the Propeller communicate (read/write) with SDHC cards.
I haven't used this particular adapter myself(at least I don't recall doing so) but I've used several others that are very similar with the Propeller and SDHC cards.
Yes, it is. There is a wrinkle in that technically the uSD standard doesn't require support for the SPI mode which Propeller objects use, but I don't think anybody has found a uSD card that doesn't support it, probably because uSD and regular SD use the same controller chips.
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I haven't used this particular adapter myself(at least I don't recall doing so) but I've used several others that are very similar with the Propeller and SDHC cards.