Speech discrimination from animal sound
The essence of your problem is one of Voice Activity Detection (VAD). This is a well researched topic and you should investigate the available algorithms - there are many papers available online, here for example

You should apply a VAD algorithm to the signal and replace non-speech areas with silence, you can then look at the specific edge cases which are concerning you, such as animal sounds have being incorrectly considered speech.

For the case of animal sounds you may want to investigate Formant Detection (for example this paper) since as you pointed out there is a distinction between them. You may also want to consider developing a heuristic based on your own empirical data about the differences. You can take many signal features such as the fundamental frequency, the signal power/amplitude, peakedness etc... and weight these appropriately to give you a confidence rating about the likelyhood of it being animal rather than human speech. This is a fuzzy approach, you would need to iterate a lot on the weightings and metrics.