Hi and welcome to the forum!
For the first problem, check your DMA settings. You want your writer to be using UDMA not PIO mode. Instrucitions are here :
The second problem may be caused by running certain region protection software programs in the background. These programs work by changing the data read from the writer on the fly so as to remove region protection. Unfortunately, when you are using something like CDCheck, changing the data read from the writer means that it will not match the checksum data it is being compared again. Disable such region protection software during burning and comparing of data using CDCheck.
I am not 100% sure about the third problem. The DMA settings for the first problem might cure it but it may also be down to the type of CD-Rs you are using. It is possible that the writer does not support these at the highest burning speed the writer can manage for CD-Rs (40x).
You might also want to update Nero to the latest version for 6.x.x. which is 126.96.36.199. Don't bother upgrading to Nero 7 - it requires a new license and is somewhat buggy besides. See here :