Hi, I can only speak to the LAX time. If you are on one ticket they will take care of you if you miss the flight. However....
I always try to allow for 3 or 4 hours in LAX between flights. You end up in queue after queue.
First is the queue for passport control, usually pretty quick, then you pick up your bag, then you have the bag check queue. I think I have waited up to 30 minutes here (or maybe it just felt like it). Then you hand your luggage to the person outside to check it in to the domestic flight. Then if you are flying economy you have to queue to get thru security into the domestic terminal. These lines are almost always really long. I am not sure how long they take, But if they were less than 30 minutes I would be surprised. So minimum time I would allow is 1.5 hours, so with 2 hours you have to hope your plane was on-time or that the airport fairy was sprinkling good luck on you.
These days I try to fly P.E. which gets me into the priority queue to go thru the OTT security in the USA.