The "properties" you mentioned
- $\infty+1=\infty$
- $k\cdot\infty=\infty$ (for $k>0$)
are called arithmetic operations for the extended real numbers. They are true by definition. With such definition, lots of theorems in real analysis can be stated in a neat way. If one is talking about the set $\overline{\mathbb R}$ of extended real numbers, then there two different "infinities": $\pm\infty$. Note that $\overline{\mathbb R}$ is defined as
$$
\overline{\mathbb R}=\{\mathbb R\}\cup\{-\infty,+\infty\}
$$
where ${\mathbb R}$ denotes the set of real numbers. In this case, the plus sign in $+\infty$ is usually not omitted. On the other hand, one could also talk about the extended nonnegative real numbers $[0,\infty]$, which appears a lot especially in integration theory.
However, one should be careful that the arithmetic operations of ${\mathbb R}$ can be only partially extended to $\overline{\mathbb R}$ or $[0,\infty]$. For instance $\frac{1}{0}$ is not defined in $\overline{\mathbb R}$: it is neither $-\infty$ nor $+\infty$. On the other hand, $\frac{1}{0}=\infty$ (as well as $\frac{1}{\infty}=0$) is used in the statement of the Cauchy-Hadamard Theorem since with such definition, the radius of convergence of any power series $\sum a_nz^n$ can be written as
$$
R=\frac{1}{\limsup_{n}|a_n|^{1/n}}.
$$
See also a discussion on the extended reals in this set of lecture notes by Terry Tao. Here is an excerpt:
Most of the laws of algebra for addition, multiplication, and order continue to hold in this extended number system; for instance addition and multiplication are commutative and associative, with the latter distributing over the former, and an order relation ${x \leq y}$ is preserved under addition or multiplication of both sides of that relation by the same quantity. However, we caution that the laws of cancellation do not apply once some of the variables are allowed to be infinite; for instance, we cannot deduce ${x=y}$ from ${+\infty+x=+\infty+y}$ or from ${+\infty \cdot x = +\infty \cdot y}$. This is related to the fact that the forms ${+\infty - +\infty}$ and ${+\infty/+\infty}$ are indeterminate (one cannot assign a value to them without breaking a lot of the rules of algebra). A general rule of thumb is that if one wishes to use cancellation (or proxies for cancellation, such as subtraction or division), this is only safe if one can guarantee that all quantities involved are finite (and in the case of multiplicative cancellation, the quantity being cancelled also needs to be non-zero, of course). However, as long as one avoids using cancellation and works exclusively with non-negative quantities, there is little danger in working in the extended real number system.
