No. The secret can be longer than the sum of all messages.

This sounds less like a one-time-pad and more like a random string broken off into multiple individual one-time-pads.

No. It's just an offset inside the key. Once a message is sent, the offset is increased of the size of the message.

Both parties can deduced that offset from the messages they perform.

Before reaching the end of that key, they can agree on another source for a new key to continue the communication.