1- If a given program runs on a processor with a higher frequency, does it imply that the processor always executes more instructions per second ?
2- If a processor executes more of a given program's instructions per second, does it imply that the processor always finishes the program faster ?
If a given program runs on a processor with a higher frequency, does it imply that the processor always executes more instructions per second ?
Generally, yes. Clock speed mostly relates to the number of instructions that can be performed per second. Always? No. (The Itanium, for example executes 3 per cycle.)
If a processor executes more of a given program's instructions per second, does it imply that the processor always finishes the program faster?
In general, yes. In the real world... consider:
There are hundreds more ways that the one processor could be faster than another. But assuming that all other factors are equal, if you raise the clock speed on a processor it will usually finish a program more quickly.1
1More quickly up to the point where the clock is in excess of the physical limitations of the chip or quantum physics, whichever comes first.