"When you keep on multiplying 1's such as, 1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity?"
In general, anything with "..." is imprecise, and it is meaningless if you cannot make it precise. I am sure that in your case it is meaningless, because...
"If so can we say this product is indeterminate?"
No. Your use of the term "indeterminate" reveals that you conflated values and expressions. For example, 2×3 = 6, but "2×3" ≠ "6". See this post for more.
"Then what about the product 2×2×2....... because this also can be written in terms of 1×1×1×....?"
Wrong. You haven't even defined "2×2×2......." precisely, and if you could then you would not be saying that it "can be written in terms of 1×1×1×....?".
"But when you consider the irrational number e , by its definition we have ,e = 1+1/(1!)+1/(2!)+1/(3!)+... .Here the right side is a sum of infinite number of rationals . What we have to say, is this because of infinity number of rationals involved ?"
No. Again. You shouldn't be saying that something happens because of "infinity [sic] number of rationals". You should instead be telling yourself that you cannot in general say that an infinite sum has the same properties as its partial sums.
Consider the geometric sequence 1, 1/2, 1/4, ... For every k∈ℕ, the sum of the first k terms is less than 2. But the sum of all of them (which needs to be defined as a limit) is not less than 2.
"We all know tan(π/2) can not be defined and it is not infinity."
No. You are free to define tan(π/2) to be whatever you like. Whether or not your definition is useful is another matter, but it's false to say it cannot be defined. In fact, it may be convenient in some (not too common) situations to define tan(π/2) to be the infinity of the projectively extended real line, but one has to learn basic real analysis before looking at extensions of the standard real line.
"But in definite integrals as an example, integral of 1/(1+x²) from x = 0 to x = ∞ We write tan inverse of infinity as π/2."
In proper mathematics, we don't write whatever we like. We give precise definitions and prove precise theorems about the objects that we have defined. It doesn't matter if we define tan to have domain excluding odd multiples of π/2. We are still free to define atan on the affinely extended real line such that atan(∞) = π/2. What properties we get for atan is another matter. Also, in basic real analysis we typically define integrals on a bounded interval, and define an integral on [0,∞) to be the limit of the integral on [0,x] as x → ∞. So in fact whether you define atan(∞) or not is irrelevant!
"I know many students don't think about these issues if we do not mention, but it is much better if we have the most appropriate way of describing these issues to give much confidence of what students learn."
If you first correct all your own errors in your understanding of the concepts, then you would be much better positioned to teach students and recognize when they make a conceptual mistake.
Understanding comes before teaching.