1) Always false. A graph with no cycles (also known as a tree) has n nodes and n−1 edges. Add one more edge and you're going to make a cycle somewhere.
2) Not necessarily true. Easy counterexample: a graph with n nodes and n edges that forms a circle (i.e. a single cycle). Take out an edge anywhere and the graph is still connected.
3) Always true, see (1). If "G has no cycle" is always false, then it always has at least one cycle.
4) Always true. Think about it like this: if it's a connected graph with n vertices and n edges, and you remove one edge, then you have n−1 edges. If it's still connected, then it's a tree. (If it is not connected, then we're done.) Now you have a tree. Remove any edge from a tree, and your tree is split into two connected components.