Why and when is lysine charged
At pH 7 lysine has a net charge of very close to +1. 

The carboxylic acid group is fully deprotonated (-1 charge).

The alpha amino group is about 99% protonated (+1 charge).

The side chain amino group is fully protonated (+1 charge).  

You need to look at the pK of each ionizable group, consider what fraction of the group is in each ionization state (for example using the Henderson Hasselbach equation), and add the charges of each group to get the net charge.  

For isoleucin, at pH 7, the net charge will be almost neutral, because 7 is well above the carboxylic acid pK of 2.32 and significantly below the amino pK of 9.76.  There will be a small fractional negative charge because 7 is closer to the pK of 9.76 than to the pK of 2.32.  Note in the first linked table in the question, there is a column for "pI".  The pI will be the only pH where the amino acid has a net neutral charge.