Whether an axis must start at zero depends on what the mark encodes.
A gap in the data is information.
Never draw a line across missing values. Leave a gap in the line, and say what is missing in the subtitle or an annotation. A line through a gap tells the reader a value exists where none was recorded. Treat missing as missing, not as zero.
Enough to read values against, not so many that they compete with the data.
Put time across and long names down.
Time runs along the horizontal axis, left to right (right to left in a right-to-left layout).
When category names are long, switch to horizontal bars so the labels sit on the vertical axis and read normally. Do not rotate labels: rotated text is slow to read and still overlaps.
Short, localized, and without repeating what the reader already knows.
For data that spans several orders of magnitude, and for readers who expect one.
A log scale shows rates of change and keeps small values visible next to large ones. Many readers misread it as linear, so use it only for an audience that works with log scales, and label it: "Cases (log scale)". Never draw bars on a log axis, since they have no zero to start from.
Two scales on one chart let you draw any relationship you like.
Use a second vertical axis only when both scales are synchronized (the same zero, and gridlines that line up) and each axis is labeled with its series and unit. Otherwise draw two charts, one above the other, sharing the horizontal axis. See Choosing a chart for the alternatives.