sooner - rather

I'd sooner do X than Y - I'd be quicker to do X than Y (I'd hesitate less about doing X than about doing Y),

I'd as soon do X as Y - I'd do X as quickly as I'd do Y (I would no more hesitate about doing X than I'd hesitate about doing Y).

This one: I would sooner do X than do Y - I would prefer to do X than Y.

The similar one: I would as soon do X as Y - I'd be just as happy to do X as Y.

I'd as soon stay at home as go to the match - I think the expression is often used to express a very slight, and maybe surprising, preference.

There is not much difference, if any, between "I'd sooner do ... than ...." and "I'd rather do ... than ...".

If the child is no better ... >> If there is no sign of improvement.
Better is a comparative in this sentence.
If the child is not better ... >> If the child has not completely recovered.
This time, better means fully recovered, not less ill.
The difference between the two sentences is mostly because of the difference in meaning of better.

She is not a beginner ... <<>

The research project involved attendance of a taught course.
The research project required attendance of a taught course.
The research project required attending a taught course.
A requirement of the research project was attending a taught course.

One-third of employees had worked less than five years.(plural)

one-third of the employees had worked less than five years. (singular)