Graphs connected, loops-free, and Multigraphs traversable I would like to know if I'm right with these exercises.
1. Which of the following graph are?

a. Connected: 1st, 3rd
I'm discarding the 2nd because even there are two arcs that intersect, I read that this doesn't mean that there is a vertex in the intersection, so I'm assuming that the arc intersecting the triangle is not connected because I can't get from one of those vertices to one in the triangle. For the same reason and discarding the 4th. Let me know if I'm wrong.
b. Free of Loops: 1st, 2nd, 3rd
c. Graphs: 1st, 2nd
I'm discarding the 3rd and the 4th because they are multigraphs.
2. Which of the multigraphs is traversable?

The multigraph that are traversable are the 1st, 3rd and 4th.
 A: In short, you are correct for 1a, 1b, 1c. Note that for question 1c we would rather say simple graph by opposition to multigraph. Indeed a multigraph is still a sort of graph. 
I cannot say for sure for 2 as it lacks the definition of traversable. My guess would be that it means the graph has an Euler path : a trail that visits every edge exactly once (allowing for revisiting vertices). It is known that a connected graph has an Euler path if and only if the number of vertices with odd degree is exactly 2 or 0. Therefore your answer for question 2 would also be correct.
For question 1.a., if you want to understand why two edges may cross not at a vertex : There is a crucial difference between a graph and the drawing of a graph. A graph is defined whithout the use of drawing. It is a set of elements (called vertices), together with a set of pair of element (the edges). When looking at a drawing of a graph, you can move freely any vertex, or bend an edge as far as you want. The different drawings you will get will always represent the same graph. Lets look at your second example. We have a graph defined on 5 vertices, we can label them $\{1,2,3,4,5\}$, and four edges $\{(12),(23),(31),(45)\}$.

This means that you can move the vertex around and get the following drawing : 

This drawing is clearly of a disconnect graph. Therefore this will be the case for any drawing of this same graph, even if you put a more complex one such as : 

This is the same graph. Different drawings but of same graph because the vertices and edges are the same. 
For a given drawing, when two edges cross (not a a vertex), this is actually called a crossing. After that we can look at some related questions, mainly : For a given graph what is the smallest possible number of crossing in any drawing of this graph. If you can find one drawing with no crossing, then the graph is called planar. This is the case for instance of you second example. Even if the drawing you have shows one crossing, the second drawing I gave you is a drawing of the same graph but with no edges crossing. Therefore it is a planar graph.
Edit As explained in the comment, the four initial graphs are all planar, the fourth one can be drawn as

