So "rebalancing" is kind of a word for a side-effect of normal routing.

It's all based on fees. So if you have a channel with "the network" (either through 1 or more channels to a big hub, or some other way), and I have a single channel with you, then another channel with "the network".

Our channel started off with .5 BTC from each of us. Then I paid you .4 BTC so the channels are:

You: .9 BTC, Me: .1 BTC.

Nodes broadcast fees, so you would broadcast that for me to send money from me to you (either directly, or through a "hop"), it will cost 20 satoshis. But you can also broadcast that for someone to send money the opposite direction (from you to me), it's free right now. Someone that wanted to pay a 3rd party unrelated to us would try to find a route that went from you to me then to the rest of the network.

So really "rebalancing" is just incentivizing someone else to rebalance your channel by exploiting their selfishness and want of low transaction fees and to include you as part of a hop. Fees can even go negative, which would make it super beneficial for someone else to include your channel in their transaction (even if it's out of the way, or doesn't serve any real purpose) to get a bit of money from it!

Who actually broadcasts the fees can be found by looking through popular node software (each node only broadcasts one direction of the channel, and I don't remember which it is right now), as well as the broadcast system (which I genuinely don't know off the top of my head).

>Aren't timelocks enforced on chain? How can we agree to extend our timelock without committing that agreement to the blockchain?

Yes and no. They are "enforced" on chain in the sense that they can't be accepted into a block until a specific block number. But they aren't ever broadcast to anyone else until you are ready to use them. In essence all LN transactions are created, but kept secret until ready to be used. The whole concept of LN relies on the fact that if we make a transaction but never broadcast it, did it really happen? If you and me agree to make a new transaction saying "this expires in 3 days from right now", technically the original transaction is now valid once 3 days pass from it, but if you try to broadcast the tx that forces all the money to you, I can broadcast the transaction that proves that we did something else after that (our second "update the locktime"), and then gives all the money to me.

Every time we do something to transact, we also create a set of transactions that serve as proof that the old transactions are in fact "old" and shouldn't be accepted, and if anyone tries to broadcast them, we can invalidate them (and enforce a "penalty" to dissuade anyone from trying to cheat).