Some cities in EU currently have pretty cheap and wide open J fares.
Many airlines have cheap fares RT, but TAP has cheap fares that allow OW bookings. OW fares are as low as $612-$900 to NYC from cities such as CPH, OSL, BUD, WAW. These fares are J class, come with one free stopover/open jaw, and earn 200% RDM on A3. You can pay an extra $150 or so for them to be fully refundable (or buy the cheaper fare and pay a similar cancellation fee). They are bookable on Amextravel.com for redemption and dates are valid until end of schedule.
If you have an option, the LIS-JFK flight features their new business class product from AZUL, which is superior to the older product on their EWR flights (but both should be lie flat)
Child fare is 75% of adult fare (+YQ).