If I have points in multiple Star Alliance member companies, can I sum up all these points to get a Star Alliance flight ticket?
No, you can not transfer points between Star Alliance carriers.
There are some mechanisms that can be used to transfer between specific carriers, such as points.com, or transferring via a hotel program (eg, carrierA -> hotel program -> carrierB), however these are generally poor value as you end up losing a significant number of points during the transfer(s).
The points can't be merged once they are earned, as Doc says. What you need to do is stop earning in different programs. When you fly any Star Alliance airline, you can credit the points to your chosen program. You don't have to choose a program that is offered by the airline you fly the most, or the airline in your own country. Choose the one that is easiest for you to earn nonflying miles with (for example I live in Canada and can get credit cards that earn Aeroplan miles, but would find it very hard to get one that earned Lufthansa miles), that has the best redemption rates (miles needed, extra fees etc) or, if you fly enough to earn status, that offers useful status benefits for you. Take a good look at the airlines you fly and decide whose program will be getting all your miles from now on.
You can also earn points on CCs such as Amex that can be converted to Aeroplan if you need it, or used for other purposes should you find that Aeroplan points can't be converted to a usable seat. Mar 22, 2014 at 16:52
If you live in the right country, @SpehroPefhany - that's harder if you live outside Canada. Mar 22, 2014 at 18:12