It is somewhat unclear from your question whether you are aware, but you cannot simply bring the baby with you. You must purchase a ticket for the lap child or infant, albeit at a reduced fare; for international flights, this is commonly 10% of the accompanying adult's fare. The infant's reservation will be provided to El Al along with the rest of the party's, and El Al, like all major airlines, understands the concept of a lap child.
I would strongly advise, however, calling both American and El Al to request bassinets; such requests are not always transmitted correctly to partners. A bassinet would make a huge difference in comfort on a transoceanic flight, and you will also be able to ask any other questions about services aboard your particular flights.
With regards to the car seat and stroller, El Al's Flying with Infants page notes
It is also advisable to bring with you a carrier and/or carriage or stroller for use at airports. The carriage will be marked at flight check-in and you can use it right up to the entrance to the aircraft. It will be returned to you as you disembark at your destination.
and further
Parents traveling with a baby are entitled to bring a fully collapsible stroller or an infant’s carry-seat or car seat, to be brought to the entrance to the plane.
Upon landing at your destination, the ground crew will bring the stroller/infant carry-seat/car seat as close as possible to the plane entranceway.
If you check your car seat, it will more or less be treated like any other bag, and unless you have a long stopover, would be tagged to your final destination. American will hand it off to El Al at your outbound connection, and vice versa on the return. You would reclaim it where and when you reclaim the rest of your luggage. As to "any issues," well, that is a bit broad. There is a risk that it could be damaged or soiled or lost, the same as anything else you check with the airline, but airlines transport innumerable infants along with their paraphernalia without incident every day.
