Well actually you aren't downloading anything. That would be getting it from the internet. In this case you are just transfering (either copying or moving) the files from one device to the other.
You, when you connect the phone to the computer, need to locate and open the correct folders where the files are kept. Just connecting it to the computer won't move the files.
Photos taken with the phone are usually in the DCM file folder or one of the sub folders under it.
To find the right folder on the phone, you will just have to look. Once you find the files you wish to put on the computer, then select all the files and I suggest you choose to 'copy' them rather than move them. While you don't have to leave them on the phone, unless you want to, doing it as a 'copy' first is much safer. For example, say there is a problem with the files when on the computer but they are fine on the phone. If you had just moved them, then you may not be able to move them back. So make a copy of the files and place that in a folder on the computer. Then make sure the files are all working correctly on the computer before removing them completely from the phone.
Then your computer may not be seeing the phone correctly. Some phones do not automatically show on the computer or show their files.
When you first connect the phone to the computer, if you get a popup on the computer screen that lets you choose how you want to access it, be sure you choose the option to view the files.
If that does not show, then once the phone is connected to the computer, tap the phone screen so that you can see the phones main screen. Tap and slide down and you may see the way it is connected to the computer, with the option to tap for more options. If that is the case, or if the phone screen automatically shows you a list of options on how you will connect, then you want to choose the option to transfer/access files.
Can't say, but you should be able to see the card from the computer as well, it would just be a different choice to view. However, if you have the card set up a different way, locked, etc., it may not show correctly on the computer.