I can sing C3 and B2 very easily, and A2 quite clearly and strongly. But I can not really hit G2 at all, just barely in a really horrible way. So within a single tone, my lower range dies completely.
Is that normal? By contrast my high range degrades over a few notes from "that's OK" to "it's in tune but it's horrible".
Yes, that's pretty normal. Lower range limits tend to be pretty hard and not impressible much by warmup or similar. However, they may vary with the time of day and other external circumstances (exerting one's voice over an evening tends to take quite more from the top than it may add at the bottom, but the morning after may show non-permanent extensions on both ends of the range).
This is entirely normal and makes sense when you look at how the voice works.
The vocal folds, commonly referred to as vocal cords vibrate to create the sound. Similar to a guitar string, the more tension on them the higher the pitch. The muscles primarily involved in this are the cricothyroid (CT) and thyroarytenoid (TA) muscles. As these muscles contract the pitch increases.
To obtain the lowest note you can sing you relax both of these muscles completely. Since they are relaxed completely the anatomy of your throat determines the lowest pitch.
At the other end, to produce your highest note you contract your CT and TA muscles as much as you can. The conditioning of the muscles and the flexibility of your vocal folds is what determines your highest pitch. These are things that can vary a lot more than the lower end so the high end is going to be a more gradual change.
Note: I found one paper (.pdf) from 1972 that makes the assertion that contraction of the sternohyoid muscles causes the pitch to lower, but I can't find any other source to support this hypothesis.
Yes, it's normal. Although it's often possible to squeeze out an extra high note or two, the lower cut-off is generally non-negotiable. Practice scales and other exercises, you might find a little extra low range. But don't be disappointed if you don't.
