1) Rather than two amino groups, you here have two ethyl groups! The 'di' refers to the ethyl. So you have an NH as your parent molecule, with a C2H5 on either side, taking the nitrogen atom's remaining bonding electrons.
You appear to have confused it with ethylenediamine aka 1,2-diaminoethane.
2) Neither of those is considered correct. You always want the lowest sum of numbers when numbering your substituents, and hence it should be 1-chloro,2-methylbut-2-ene.
3-methyl-4-chloro-but-2-ene - or indeed 2-methyl,1-chlorobut-2-ene - is undesirable as your prefixes should be ordered in alphabetical order. Thus 'chloro' comes before 'methyl', as c comes before m.
3) My understanding is that a carbon chain with attached methyl and ethyl groups would NOT be an 'unbranched chain'.
Regardless, your first step is to identify the longest carbon chain. That is indeed, a carbon chain with 5 carbons, so it is 'something' pentane. We have methyl groups attached to two different carbons, so as in question 2 we will contrive to get the lowest sum of numbers on our substituents. It will be either '3,4' or '2,3' - the latter is preferable, and hence we have 2,3-dimethylpentane.