Can't we just write a program in machine level language?

Question 1

Yes, you can program in assembler under Linux.

Check this and this questions on Stack Overflow, for example. Also, the Linux Assembly HOWTO looks good.

Question 2

Compiler convert human understandable language into machine level language. Can't we just write a program in machine level language so that it will be easy and quick for a program to execute.

No one writes programs in machine language. Normal binary exectuables are not just machine code anyway, so it would be pointless to try and do this. Binaries contain machine code but include specific, OS dependent formatting. For example, linux uses ELF. This format is understood by the linker and loader (on *nix, the loader is part of the kernel). The only place unadulterated machine code exists is in the system memory.

You can write programs in assembly language, which is very similar to machine language, but then this must be compiled and linked. In other words, it is the same thing as writing a program in any other compiled language.

Finally, creating a binary manually by formatting some machine code would not provide any advantages and it would be an endless headache to work with. You might do it as a learning excercise, but not for any real purpose.

Question 3

Yes, but then you have to write more and then it takes longer, so using a higher level language and compiler is more efficient on your time. More often than not, a compiler is better than what a mere human can do (they are pretty good programmers these compiler writers).