Dividing or subtracting : Normal PDF's? of independent random variables There is clear rule how to multiply OR sum Normal PDF's i.e.
https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
$N_1(\mu_1,\sigma^2_1) + N_2(\mu_2,\sigma^2_2) = N(\mu_1+\mu_2,\sigma^2_1+\sigma^2_2)$
Is there similar formula for DIVISION and SUBTRACTION of normal PDFs ?

Here is the product :
$N_1(\mu_1,\sigma^2_1) * N_2(\mu_2,\sigma^2_2) = N((\sigma^2_1*\mu_1+\sigma^2_2*\mu_2) / (\sigma^2_1+\sigma^2_2), 1/ (1/\sigma^2_1)+ (1/\sigma^2_2))$
sorry dont know how to format those things 
 A: Since we don't seem to be getting the message across, here's the problem with your question:

(These examples are for $X_1\sim N(0,1/4)$ and $X_2\sim N(5/2, 9/4)$. In column B the random variables were assumed to be independent.)
You need to pick whether you're really asking about column A or column B. You're currently asking about A2 and B1. They're different kinds of things!
Edit: additional explanation -- 
Let $X_1\sim N(\mu_1,\sigma^2_1)$ and $X_2\sim N(\mu_2,\sigma^2_2)$. The density of $X_1$ is $f_1(x)$ and the density of $X_2$ is $f_2(x)$.
Then the sum of the two densities is $f_1(x)+f(2)(x)$. This is shown in image A1. (It is not a density, but if scaled to integrate to 1, corresponds to a finite mixture of normals.)
The product of the two densities is $f_1(x).f_2(x)$. This is shown in image A2. (It is not a density, but if scaled to integrate to 1, corresponds to a normal.)
[Since we're dealing only with operations the density functions, independence of $X_1$ and $X_2$ is irrelevant to the above.]
The density of the sum is the density of the sum $X_1+X_2$. The case for jointly normal $(X_1,X_2)$ is given in Dilip's answer. If they're independent, that gives the result in your question. They may of course not be jointly normal, but dependent in some other way, in which case the distribution of the sum will be different. The independent case is shown in image B1.
The density of the product is the density of the product $X_1\,X_2$. The general case is a little complicated, but some special cases are more well known. None of these are normal, but some cases may be approximately normal. For example if both $\sigma_i$s are small relative to $|\mu_i|$, and the variables are independent, you get a distribution that's approximately normal. The independent case for random variates from the two normals mentioned above is shown in image B2. 

If you're interested in the difference and ratio of density functions (i.e. all column A-type calculations), here are examples using the two normal pdfs above above:

A: *

*If $X$ and $Y$ are jointly normal random variables, then $$X\pm Y \sim N\big(\mu_X \pm \mu_Y, \operatorname{var}(X)+\operatorname{var}(Y)\pm 2\operatorname{cov}(X,Y)\big).$$

*If $X$ and $Y$ are independent $N(0,\sigma^2)$ random variables, 
then $\displaystyle \frac{X}{Y}$ (and also 
$\displaystyle \frac YX$) has a standard Cauchy distribution.
Both of the above facts have been discussed numerous times on stats.SE


*

*Your assertion about the distribution of $XY$ is incorrect: $XY$ 
does not enjoy a normal distribution.

