I learned that we can define extended reals and some related arithmetic operations. I wonder when can we use these notations without worrying about consistency.
For example, if $\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=\infty$, I can verify that 1. by definition, $\infty+\infty=\infty$; 2. using $\epsilon-\delta$ method, I can prove that $\lim_{n\to\infty}a_n+b_n=\infty$. Thus in terms of limits, it's ok to apply the addition rule to even infinite limit points.
However, what about other maths concepts I had learned? Is it always true that as long as there exists a clear definition that includes infinity (such as limits of a $\bar{\mathbb{R}}$-valued function), then all rules that apply to finite numbers (such as addition of limits) also apply to infinity? If not, do I need to check every time by myself?
All of the limit laws work just as well as before. If $l$ and $m$ are extended real numbers, and $f(x)\to l$, $g(x)\to m$ as $x\to a$, then
provided that those expressions to the right of $\to$ are defined. Remember that $\infty-\infty$, $0\cdot\infty$, $\infty/\infty$, and $l/0$ are undefined.
This is no accident. In fact the reason that it makes sense to define $\infty+\infty$ as $\infty$ is that the "sum limit law" still holds under this definition. Contrariwise, since there is no way of defining $\infty-\infty$ to make the "subtraction limit law" work out, we ought to leave this expression undefined.
More deeply, one of the motivations behind the extended reals is to extend the usual arithmetic operations such as $+$ and $\cdot$ to $\pm\infty$ by continuity. The usual addition function $+:\mathbb R^2\to\mathbb R$ can be continuously extended to the point $(\infty,\infty)$, and so this is what we do.
A caveat: in measure theory and probability, I believe it is common to define $0\cdot\pm\infty=0$, even though this is not a continuous extension of multiplication. I have no doubts that this convention must be useful for other reasons, but I do not know what they are.