How does one go about measuring the amount of consolidation in a market, the competitiveness that results from that consolidation level, and the efficiency that results from that competition?
There are several measures of competition. Some common ones:
Learner Index: $L= \frac{P-MC}{P}$, where $P$ is price and $MC$ marginal costs. This measure measures competition by looking at pricing power of firms.
Herfindahl-Hirschman Index $HHI = \sum_i MS_i^2$ where $MS$ is market share. This measures competition by seeing how big market share firm have.
Of course, this is just tip of the ice berg there are many other indices but the two above are probably the most used ones in the literature. You can learn more about different empirical measures in Boone (2008) which has good overview, although depending on your level it might be bit technical.
When it comes to market efficiency there is no simple measure. Usually you will have to do some modeling of counterfactuals under various different scenario and inefficiency would be measured as deviation from the best scenario (scenario that delivers highest Marshallian welfare).
PS: Your question has several false statements:
in other words, the majority of what we're calling inflation is simply price hiking or price fixing
price fixing is almost everywhere illegal. Also regardless of the source of inflation, inflation occurs when average prices rise regardless of what the reason was for it, so whether price increase because of market power or price fixing or monetary policy it is still inflation. However, you should also note there is always a maximum limit to what price would be on the market (e.g. monopoly pricing) and even with price fixing/market power price would not increase beyond that so if you see prices increasing over long period of time it almost certainly cannot be attributed to market power/collusion as even monopolist have a limit to which they can raise prices to maximize profit, since beyond certain point demand for their product falls too much.
Monopoly is illegal but what about oligopoly?
Monopolies are actually not illegal in most countries. They are regulated but I do not know many countries where they are outright illegal. Oligopoly is also legal in most countries.
competition creates efficiency
This is actually not always true. A monopoly can be more efficient than a market full of perfectly competitive firms. For example, contestable natural monopoly will be more efficient then if in the same market there was competition.
