What is "a point at infinity" in the complex field?
What is the neighbourhood of a point at infinity?
What does it mean if a connected set contains "a point at infinity"?
If I understand correctly, you're asking what $\displaystyle \mathbb C \cup \{ \infty\}$ is (called the Riemann sphere). As a set it's simply the set of complex numbers $\displaystyle \mathbb C$ together with an additional point you call "infinity" and denote by $\displaystyle \infty$. As a topological space, it's the one-point compactification of the complex plane: a set is open in $\displaystyle \mathbb C \cup \{ \infty\}$ iff it's open in $\displaystyle \mathbb C$ or is the complement of a compact set (in $\displaystyle \mathbb C$).
This definition should be enough for you to answer the two other questions.
Intuitively, imagine you grab the real line and wrap it around a circle. It's "natural" to say the line won't cover the whole circle: there will be one point missing, so you call this point infinity. Same goes for the complex plane folded around a sphere: one point of the sphere is missing, so you just cover the hole with an extra "point at infinity". That's why it's called the Riemann sphere.
Hope it helped
By "the complement of a compact set (in $\displaystyle \mathbb C$)" I mean the complement (in $\displaystyle \mathbb C \cup \{ \infty\}$) of a compact (in $\displaystyle \mathbb C$). For instance:
$\displaystyle \{z\in\mathbb C:|z|>2\}\cup\{\infty\}$
is an open set in $\displaystyle \mathbb C\cup\{\infty\}$ since its complement is the closed ball of radius 2 centered at the origin, which is compact.
Not exactly relatively open (although you can think of it like so in practice). It's connected if it cannot be written as the union of two nonempty disjoint open sets in this new topology we defined. For instance, my previous example:
$\displaystyle \{z\in\mathbb C:|z|>2\}\cup\{\infty\}$
is connected in $\displaystyle \mathbb C\cup \{\infty\}$. But not in $\displaystyle \mathbb C$: it's not even a subset of $\displaystyle \mathbb C$! In practice, though, one can "ignore" the point at infinity (for connectedness related issues) and just think of it as a set in $\displaystyle \mathbb C$, in which case your description is, I believe, valid.