First off, let me say that you can find the answer to this question in Sage using the nauty generator. If you're going to be a serious graph theory student, Sage could be very helpful.
count = 0
for g in graphs.nauty_geng("20 180:180"):
count = count+1
print count
The answer is 4613. But, this isn't easy to see without a computer program.
At this point, perhaps it would be good to start by thinking in terms of of the number of connected graphs with at most 10 edges. Then, all the graphs you are looking for will be unions of these. You should be able to figure out these smaller cases. If any are too hard for you, these are more likely to be in some table somewhere, so you can look them up.
Connected graphs of order n and k edges is:
n = 1, k = 0: 1
n = 2, k = 1: 1
n = 3, k = 2: 1
n = 3, k = 3: 1
n = 4, k = 3: 2
n = 4, k = 4: 2
n = 4, k = 5: 1
n = 4, k = 6: 1
n = 5, k = 4: 3
n = 5, k = 5: 5
n = 5, k = 6: 5
n = 5, k = 7: 4
n = 5, k = 8: 2
n = 5, k = 9: 1
n = 5, k = 10: 1
.
.
.
n = 10, k = 9: 106
n = 10, k = 10: 657
n = 11, k = 10: 235
I used Sage for the last 3, I admit. But, I do know that the Atlas of Graphs contains all of these except for the last one, on P7.
Yes, there are. I'll describe two such graphs.
First, arrange the six vertices in a 2 by 3 grid. Then connect vertices so as to form the number $8$ as seen on sports scoreboards or some digital clocks. This is what a commenter refers to as a theta graph.
For the second example, call the vertices of degree $3$ $A$ and $B$ and the other four $x,y,z,w$. Set $A$ adjacent to $x,y,z$, $B$ adjacent to $x,y,w$, and $z$ adjacent to $w$.
In the first example, the degree $3$ vertices are adjacent but in the second they are not, so the two graphs are non-isomorphic.
Best Answer
Just consider $E_3,P_3,C_3$, and a graph with $V=\left\{v_1,v_2,v_3\right\}$ and only one edge, for instance, $\overline{v_1v_2}$.
If you're unfamiliar with the notations above, they refer to the empty, path, and cycle graphs of order 3 respectively.