Does the contract curve always have to connect the initial points on an edgeworth box? Why or why not? What if the contract curve goes out of bounds? In that case, do I assume it superimposes itself on the axes it is closest to?

I hope this was clear. Not sure how to explain this other than visually...
The contract curve is the locus of Pareto optimal points in an Edgeworth box. What we get from that: 


To be P.O., an allocation must be feasible. So, the contract curve does not extend beyond the edges of the box (opposed to indifference curves, which we can draw as extending beyond the edges of the box) because points outside the Edgeworth box are not feasible. 
Because the Edgeworth box does not allow for destruction/disposal, we know that the initial allocation point will account for all resources in the economy. This may make you think that the initial endowment should be part of the contract curve. However, this is not true. That is, the initial endowment point is not necessarily part of the contract curve because there are easily imagined cases where trading happens without making either agent worse off and makes at least one agent strictly better off.  
Yes, there are certain circumstances that cause the edges of the box to form part or all of the contract curve. If you want to explore the idea more I suggest you spend some time reading about what happens when one agent has (or both agents have) linear indifference curves.