I do it that way, our professional photog friend does it that way, unless she is reading off of cards that are not in the camera, so do 100's of 1000's of others do it that way. It is not going to screw things up, reading the card while it is in the device.
I don't know if it is safer but I always leave the card in and read from the camera. I have never had a problem downloading to any machine. I can erase the card when done downloading from the PC if I want.
Not sure why your drive is not recognizing the card but suggest a driver error as well. I find the card readers to be kind of fragile.
Please do not be insulted but I am famous for sometimes ignoring the obvious. Is your card compatible with the reader/drive.
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Safer, since if you are just a casual photog, you leave in. Even professionals are now going for the largest SDHC cards, so that they do not have to swap out cards all of the time, and possibly loose one. I just replaced our 2gb with a 8gig, and this past Sunday, still had tons of space after taking about 115 photos on the highest res. Ended up using about 420mb.