How relevant is the concept of ‘class’ to research in politics or political economy today?

Answer 1

Not very. The notion of class has been supplanted by those of income percentile and wealth percentile. Present-day researchers are more likely to study "the third income quintile" (translation: the middle 20% of households in the income distribution) or "the 99th wealth percentile" (translation: 1% of wealthiest households) rather than "middle class" or "super-rich".

Answer 2

It is very relevant, although classes are now defined by level of income rather than ancestry or occupation. I.e. the consumption patterns of middle-class people are very different from poor people.

Of course, having middle-class income is kinda hard with certain jobs and low levels of education, but not completely impossible. Likewise, eduction or job do not necessarily guarantee and income.