The grammatical gender of a child is 'it'. Please note there is not always a one to one relation between grammatical gender and biological gender. For example, large vessels are referred to as a she whike you woukd expect it. Or a mouse, insect etc is an it (even though it can be a he or a she).
If by "recent years" you mean a few centuries then I agree. It was common even as early as in the 16./17. century and some even claim the singular "they" was around since the 1300s (although I haven't had the time to verify it myself yet, so I don't know how valid those claims are.) Only later grammarian prescriptivists started to have a problem with singular "they" and only now it's coming back to graces. (Just like grammarians also used to have a problem with - also perfectly valid - finishing sentences with prepositions)
Sorry, folks, but to this Old Fart (not to panic, those of you of tender sensibilities, the term is a very, very mildly risque euphemism for one of advanced years), the general acceptance of the singular “their” is proof positive of advancing entropy. Abandon all hope; surely, The End is near; Doomsday is rapidly approaching!!!
Your example is perfectly acceptable. I faced a similar problem some time ago, writing text for the UI of a social networking website. I needed to write things like "Your friend would like to share their pictures with you - accept?". That sort of thing appeared everywhere throughout the site, and the alternative of using the clumsy "he or she" and "his or her" all over the place was a non-starter :)
An answer to this question has to start with realizing the quite odd meaning of this sentence: the child has for some reason inserted its finger into an unidentified mouth that is, implicitly, not his or her own, but, yet, the owner or nature of that mouth (is it the mouth of a volcano or a person?) is remaining consciously unidentified.
There are times when translating between languages that context has to either be inserted or removed to yield a natural result in the target language [just like the explicit possessives have to be removed translating the English version of this sentence to Russian]. Not a native Russian speaker, but I think this would be one of those cases: something about which mouth this finger has wound up in would be included. "The child has its finger in someone's mouth," or what have you.
Textbooks seem to say something like во is used "before difficult consonant clusters" and sort of leave it at that. Probably b/c the matter is a touch complicated. I found this more comprehensive listing: http://ask.masterrussian.com/279/what-is-the-rule-for-using-%D0%B2%D0%BE-%D0%BA%D0%BE-%D1%81%D0%BE-instead-of-%D0%B2-%D0%BA-%D1%81
A person's own body parts (or relatives, as another important example) are not prefaced with possessives in Russian unless for some reason there would be a lack of clarity about whose body parts are meant.
у + person in genitive is a structure that is used to indicate possession in certain cases, particularly when that possession is extremely "deeply rooted," like parts of one's body, places one lives.
Might I suggest 'their'? Though historically only used for third person plural, it has become the most common word to refer to a third person of unknown gender, or a hypothetical third person.
'Its' is usually reserved for non-human animals of unknown gender or inanimate objects.
Nope, it is "where at". Some consonant-ending masculine nouns (ad a few nouns like кровь, ось, дверь) are just like this.
That -у-ending is always stressed, and it is not optional for common nouns (i.e. native speakers consistently say в году, на полу, во рту instead of using the normal ending -е)
At times some languages get into complications, where others make it simplier: here is the need for English to add a possessive pronoun whith inherent problems; simpler ,,even if more undeterminate in Russian (and Italian, too: " il bambino ha il /un dito in bocca: the child has the/a finger in mouth); but in Italian "bambino" is masculine; so even in this case a choice would be necessary