2-clause satisfiability associated graph
You already have a description of how the graph is built from the formula. Think about a possible algorithm for doing the job. For example, you first scan over all variables in the formula, and you add two vertices into your graph for each. Now you only need to add in the edges into the graph. You can scan over the clauses, and if you see either form you describe, you add in an edge into your graph. This all can be implemented to run in polynomial time; just observe you are performing two scans over the formula, and you do a bit of work for every step of the scan. Similar reasoning will make you see the graph only needs polynomial space.

For your first question, an answer is given in the Wikipedia article on 2-SAT, but has also been touched upon in other question (see e.g., here).