Research Questions - General Questions about Specific Situations

Good Afternoon Community Members,

One of the reasons I joined this community recently is to improve the way I perform my research, which I am happy to say has improved since I joined. Yet, I still have questions which might be answered by the collective wisdom of the community. Some of these may have been asked and already asnwered, but I ask in ignornace.

Here are my questions:

In researching christening records, I have noticed the occasional listing with only a mother's name listed/captured. I would think this is obviously a situation where a child is being born out of wedlock or some other related situation. However, I noticed yesterday, the same mother had 3 different kids over a period of 7 or 8 years with no father named (No relation to me).

Is there something else at work here or is it just a situation where a woman is having children out of wedlock? Is there something else we can gleam from seeing this trend? Thoughts/Opinions???

In working with the familysearch.org website, I have noticed lots of indexes where a film number is captured, but no image is available.

Does this mean the image is available through some other resource, such as through one of the LDS Family Research centers?

What is the relative research value of familysearch.org vs. an LDS Family Research Center? Is more information/resources available though one of the research centers vs the website?

Thank you, in advance, for your thoughts and input.