The verbs of perception take either the gerund or the infinitive: see, smell, hear etc. However, the gerund is more commonly used. It sounds odd to simply hear the infinitive. Technically, the gerund takes a possessive. I can hear his singing in the shower. However, this rule is slowly falling out of use, unfortunately.

They are both ok. The meaning is essentially the same, however there may be slight nuances between the two. Using "singing" may communicate more frequent singing around the house where as "sing" may communicate singing on specific occasions.

Both are OK. They differ slightly different because "singing" has a continuous feel to it. If you paused and listened for a while to your neighbor, then "singing" is a better word. "Singing" in your sentence is used as an adjective, not a gerund (noun). It does not take a possessive because it is not a noun. "Gerund" would describe the situation where "singing" is used as a noun, for example "I heard the singing of my neighbor". In that situation, you have the option of using the possessive: "I heard my neighbor's singing".