The $3K JFK-LHR Z fares only have a 28-day advance purchase requirement, they also have a Sat night stay requirement (ZHXB68US fares). The other destinations have similar 28 day advance Z fares in the $5K range, but there is very limited availability for these on JFK-AMS and JFK-CDG (JFK-BRU has better availability for these). They also have 50-day advance Z fares in the $2k - 2.5K range (along with Mon-Wed outbound and Sat night stay requirements). They have better availability. All readily seen in the Google flight calendar searches below.

Summation -- JFK-LHR has pretty good $3K Z fare availability in July, but you'll need to do a Saturday night stay. The other destinations have limited availability for Z fares in July with the exception of JFK-BRU (but they are in $5K range). Beyond July, you can find good Z fare deals, but they have pretty restrictive requirements (50 day advance purchase, Mon-Wed outbound, Sat night stay).