How do I know what gear combinations should work without chain rasp on front derailleur? I tried adjusting the front derailleur on a 3x10 trekking bike. After fiddling around a long time, I concluded that is probably not possible to make all cogs work with all chainrings. How do I know what gear combination should work? Should you for example be able to use the fifth gear with all three chain rings without chain rasp? Should you also be able to switch between all chainrings in the fifth gear? 

Currently, the three largest cogs are only working with the smallest chainring. If I adjust the front derailleur to allow shifting on the second chainring, I lose the ability to downshift from the third to the second chainring in higher gears (the chain rattles but does not come down).

I also experienced something similar when I crosschain on a 3x8 city bike. I have never had any issues with a 3x7.
There are two approaches here:


The standard rule is to avoid cross-chaining. It means that you don't use large with large and small with small. The large sprocket at the rear should not be used with the large ring at the front. In the same manner you should not use the smallest rear sprocket when your chain is on the smallest front ring. In both cases the chain runs quite diagonally or sideways when you look from above. It puts a lot of strain on the drive components and the chain is most likely to rub against the front derailleur plates. With a triple the use of the middle ring is less critical. It may work with any rear sprocket. But still, depending on the position of the chain, rubbing against either FD-plates is possible, so you should avoid the use of the two extreme sprockets.
The previous being quite old school, newer derailleurs and narrow chains are less affected by cross-chaining, if you leave out the higher wear of the drive train. Especially modern indexed front derailleurs don't have just three positions but two for each chainring or even three for the middle one. You can notice these if you carefully actuate the shifting. You will notice that when the derailleur is as far right as possible, a slight touch of the lever will have the FD move inboard ever so slightly without throwing the chain off. This position is meant to be used with the larger rear sprockets. It will only work with carefully adjusted indexing of the derailleur which may be fiddly. Some makes, like SRAM have FDs with a special geometry of the plates that doesn't require trimming where electronic shifters trim the position of the FD according to the position of the rear derailleur


Note: in the days of non-indexed shifting and down-tube levers it was customary to the adjust the position of the left lever slightly when the 'ear-meter' registered noise from the chainrings.