If we are late on the LSL and the connection is very close, say 15 minutes, will Amtrak attempt to get us from the LSL to the EB? From what I have learned here the LSL and EB board and unboard in different areas of CUS.
Yes, they will.
Chicago is essentially two stub end terminals back to back. The Lake Shore, and indeed all long distance trains except the Builder, use the South Concourse. The Builder, the Hiawathas and now the Borealis use the North Concourse.
With very close connections going the other direction, off Builder to another train, Amtrak has always escorted me and other connecting passengers directly to the train on the other side, around the east end of the concourse building beside the through track (track 28/19), not going into the station at all. Sometimes this was done at a brisk trot, though.
The Builder almost always boards on Track 19, btw.
A note on hotels. While Amtrak will provide accommodations themselves on a missed connection, they will
not reimburse you for other expenses such a missed prepaid hotel reservation. They provide a ticket on another train, hotels to lay you over to catch that train if required, money for cab fare to get to and from that hotel and meals during the wait. That's it. You might wangle a substantial voucher for future travel from Customer Relations after the fact, though.
I would not worry too much about it, though. LSL to EB a pretty reliable connection, I do it myself regularly. EB to LSL, not so much.