The primary tool to manage inflation is interest rates, however interest rate movements also have other effects, positive and negative, so it's not as simple as turning on a tap.
It's important not to let inflation get too high because that makes new investment unattractive, the cost of living go up, and retirement savings worth less.
If I'm not sure what inflation will be day-to-day, I can't make an accurate prediction as to whether I should save my money or spend it.
But a bit of low level inflation is generally seen as a good thing. The questions over the efficacy of monetary policy as a stimulus tool in all situations is certainly questionable though.
It's also worth noting that independent central banks are a fairly new thing. Inflation levels used to be set by politicians who would notoriously let short term interest over ride price stability.
The Bundesbank was the first independent bank drawing lessons from the horrors of hyper-inflation. Most (all?) developed countries now follow that model.
As to the cost of living going up: in nominal terms, sure. But it depends on the type of inflation. Usually wages grow to keep up with inflation, so in the end, it's a wash.
I'm sure there are costs associated with extremely high inflation, e.g. restaurants having to print new menus all the time. However, I'm not aware of any serious work being done to quantify that effect. It seems that even at rates of inflation that we would consider very high - let's say 15% anually - those costs would still be irrelevant.
Volatile inflation would probably hurt more than merely high inflation as long as the latter is stable.
If inflation is much higher in one country than another, then exports from the country with lower inflation will be more competitive on the global market.
Inflation also alters consumer behaviour.
If inflation is very high people tend to buy goods way in advance for fear of further price increases. This "bringing forward" of consumer spending means less will be consumed in the months ahead.
The general uncertainty created by very high inflation undermines business confidence and reduces investment. Basically, you're money is better off in another country where inflation is lower.
Depends on what the inflation actually does. For global trade, it's only relative prices that matter. If relative prices did not change in the inflation, the exchange rate would adjust and that's all that would happen. This is because exchange markets react much faster (orders of magnitude) than export/import markets.
If the inflation reduces the real wages of workers in the country, then high inflation actually means that the country's exports will become more competitive. (But the workers in that country might not like it... so much for the benefits of "competitiveness").
As to the question about international investments, let me put the caveat that most of what people talk in that arena is nonsense. That said, even accepting some of the rather absurd premises: it is still a question of relative prices. If you have 0% inflation domestically, but also 0% interest, then perhaps you may want to put that money in an account abroad where you have 5% inflation but get 8% interest.
Again, it is not the high inflation itself that makes investment unattractive. Volatile inflation is another beast, but stable and high inflation has so far not been shown to be a problem.
Also note that in your last paragraph, you simply rephrased your original claim without giving an argument.