Is asking questions with very short answers on major exams recommended? As a part of TA work, I need to give questions for major exams. I need to evaluate the answer sheets of students later.

Let us consider the following simple examples of questions:

1)  Briefly describe natural numbers.

The answer to this question is not unique. It depends totally on the student's perspective on natural numbers. And the answer is generally long.

2) How many natural numbers are there in total? Is 6.6 a natural number?

The answer to this question is too short (Infinite, No). The answer does not depend on the student's perspective on natural numbers. It tests the knowledge of students regarding natural numbers.

I want to give questions of second type, which can test the student's knowledge and intelligence in detail and avoids long answers, ambiguous answers, long evaluation time, partial marking, unnecessary bargaining for marks by students, etc.

But some of my colleagues and faculty are saying that it is similar to multiple choice questions and fill in the blank questions, which are not recommended for major exams, which need long answers. I am confused, but strongly want to give answers of type 2 (single line answers). 

Assume that there is no copying during exams.
There's a bit of a false dichotomy in the way you present things. Exam questions can have a long answer without being hopelessly vague and subjective. In college-level math, one usually can't avoid asking some questions with long answers, whether they're problem-solving, proof-based, or conceptual. Partial credit is just part of the deal, as is handling the occasional student seeking more partial credit. (You have to at least hear them out, because it's always possible you've made a mistake.)

However, there are different views on this; I once had a professor who gave an entirely true/false final exam, with no explanations or partial credit, in an abstract algebra class. His reasoning was that over an exam with say 40 questions, the effects of random guessing would average out to a roughly equal small bump for everyone. I was skeptical, but after taking the test I was confident that the same people did well on it, as would have done well on a traditional "solve this problem, prove this small fact, etc." exam. But this may have worked because abstract algebra has a lot of questions with a concise statement and answer, but that require real understanding. (One T/F question I remember: The Galois group of the fifth cyclotomic field is cyclic.) I don't think it would work for every course; at some point in your math education, someone needs to read the gory details of your solutions and give you feedback.

By all means, look for opportunities to make good questions that can be graded quickly and objectively, but a lot of the time, there just aren't many questions like that.