Yes they can do so, and the indications are that NYCFC fully intends to have a proper academy which it will use to populate some of its squad. However, it's not quite as simple as it is in Europe. For a start, to sign a youth player you have to go through MLS still, asking MLS to give them a contract and then sign them through the allocation process. In order to avoid other teams stealing your youths you have to designate them one of your 'discovery' picks on them. Thing is, you only get 10 discoveries at a time, and that includes players you have "protected" for the future, even if you don't intend to sign them straight away. Importantly, this even applies to players coming out of your own academy. This may seem anathema, but you have to remember that MLS is set up specifically to be the epitomy of an anti-monopoly league - this even extends to stopping teams from being able to monopolise recruiting young players via a source no other team can match.
As for domestic and foreign players - MLS allows each team to have a quota of 8 foreign players. However, as with virtually everything in MLS, it also allows teams to trade these "international slots" to other teams for players/allocation money/Draft picks/etc. Therefore it is possible to have more than 8 foreign players but the league as a whole can never have more than 8 x (total # clubs).