I think there are also expedited security lines at TLV for business class passengers
There is no fast security line for BA passengers. Though the line for BA passengers is on the left hand side of the terminal and is less busy and moves quicker than the right hand side. (ElAl etc)
Will she be able to access the lounge at TLV using the business class boarding pass for the LHR-LAX leg of our trip later that day?
At check in they will issue the lounge invitation for business class passengers. Ask politely if they could issue for all 3 and they usually just phone the lounge to check it is not too busy and often oblige. This would be because you are entitled to 2 passes for your business seats and not because of onward travel.
if I book my wife in an economy seat, will she be able to check in with us using the business class line at the airport?
I can't see this is a problem as there is rarely a queue at the business check in desk and (particularly as this is TLV) a group travelling together would have had to just go through security screening together so would be expected to stay together.