The problem with NH2(-) as a nucleophile is that it is extremely basic and often performs side reactions.

It is more difficult for triple bonds to undergo reaction with strong acids such as H-Cl since it would result in a vinyl cation, which, being sp-hybridized, is a less stable carbocation than alkyl carbocations. Essentially you want to avoid having empty orbitals with a lot of s-character.

Ethene can be a nucleophile, yes. It can react with dihalogens such as Cl2, Br2, and I2 for example.

2.)When going down a group it is not so simple. Nucleophilicity does not necessarily mirror basicity. This idea stems from the fact that when we consider an negatively charged ion as a nucleophile we must take into consideration the type of solvent used.If a polar protic solvent is used, the nucleophile will be hindered from being able to share its electrons in attacking an electrophile because the protic solvent will hydrogen bond and form a solvation shell around the nucleophile. Basically, this means that larger anions, which are more polarizable, will be more nucleophilic because their electrons will be less hindered when compared to smaller anions in their group.Thus, in this case nucleophilicity increases as you go down a group, not up a group.

3.)However, if you use a polar apriotic solvent on a negatively charged ion no hydrogen bonding occurs.Thus, no solvation shell. In consequence, you will have nucleophilicty following basicity up or down the column(Note: I have read that even then this is not always the case.Could you elaborate?)

4.)When it concerns nucleophiles that are uncharged size dicatates nucleophilicity.The larger the atom the greater a nucleophile it is.
5.) Sterically hindered nucleophiles react more slowly.

Nucleophilicity is solvent-dependent. Polar protic solvents can hydrogen bond with halides. F- is most basic, but also forms strongest hydrogen bonds, and will carry around with it a shell of solvent molecules which hinder nucleophilicity. That is why F- is the worst nucleophile among halides in polar protic solvents.

In polar aprotic solvents trend is reversed; F- is best nucleophile, I- worst. See this post on solvents:
https://www.masterorganicchemistry.com/2012/04/27/polar-protic-polar-aprotic-nonpolar-all-about-solvents/