Hello!
FIrst off, welcome to the site and congratulations on a lovely starter pet! :D
As for your questions:
Your second assumption is correct. common and rare only refer to how often you find pets with those genes in the wild (although there was the theory that it means how many of those pets are on the site, but I haven't heard anything official on this and think the wilds theory much more likely). At either rate though, it has nothing to do with how easily a gene is being passed down. Which gene gets passed down is random. There is a 50/50 chance that the red will be passed down or the silver.
Since it ties into this question (although you didn't ask it specifically, but I didn't understand this in the beginning at all): If you hover over a gene on your pet, it says Power: XX - this power is what distinguishes which gene will be recessive and which dominant! If your mother gives silver with Power: 40, but the father passes down Red with Power 50, the child will always show Red (dominant) and only have Silver as recessive hidden in its genes. Since Silver has the weakest Power of all the colour genes, this means that you have to get Silver gene from BOTH mother and father to actually show as Silver. Any other colour passed down would always show that other colour.
Second question: The left hand column of genes in your pet is ALWAYS from the mother, the right column always from the father! It is impossible for a pet to get two genes from the same parent in one category. In your example, this means the child would get EITHER Steel OR Solid Bones and the other gene would always be Normal.