The Great Lighthouse - how do you use it?

Yep, all cities connected to your trade network will get the 2 trade route from the island city. I just tested to make sure.

Getting astronomy through liberalism sounds a bit long-winded and its probably quicker to research astro directly: avoid meditation, CS and theology. Research worker techs, sailing, masonry (for Glite), alphabet, IW, maths, MC (variable order), machinery, compass (bulb or research), single bulb optix, calendar, double bulb astro.

Coastal cities get +2 traderoutes but these traderoutes can be to any city. So if you have the GLH and only 3 cities (all coastal) and no open borders you get only one additional route per city. As soon as you have another e.g. non-coastal city connected, the coastal ones will get an additional route to that city. Of course the inland city will only have one trade route pre-currency. The additional route from currency also only applies if enough cities are connected. (I once bulbed currency with a GM from the GLH in an isolation game and wondered why no new income was generated: I did only have 4 cities, so the 4th traderoute did not apply.)

If you are isolated but have one offshore island a city on this island will give all your mainland cities one route with 2 trade instead of one because it is a different landmass.
Similarly with foreign trade.
Note that you need both open borders and access, i.e. a closed border civ or a barb coastal city can block trade access!

In strict isolation with only one landmass GLH is not so great pre-contact.
But there are some maps where you have early contact to other civs by "stepping stones" or just bigger landmasses close together you will get huge benefits quite early on with open borders. In such cases I think you can get (3x3) extra trade in your capital and typically around 6 in your other cities even pre-currency from foreign traderoutes. Note again that all this is on top of any tiles worked!

As with all wonders, on immortal (sometimes emperor) it is very possible to miss it. Although the lighthouse does not seem to go nearly as fast as Stonehenge and the Oracle because the AI has to built a lighthouse as well and even if there is a wonderwhore on the map like Ramesses he might not have a coastal capital. And there is no bonus from stone.
So around 1000 BC seems a typical date. If one is playing an IND or ORG leader one can get it a 1600 BC or so which is usually a safe margin. Of course this means that one may not even have enough cities to benefit from the extra routes.

I think the question here was about a multiplayer game on a lakes map. I don't know about the dynamics in MP and how often people sign open borders, but if they are smart, they are more reluctant to do so with you if you build GLH. Building it on a lakes map is very questionable in any case. It is unlikely that you will ever get any offshore trade routes, so even foreign trade routes are only +2c for quite a while. And you won't have many coastal cities, even if you start on the shore of a large lake.

I am not familiar with those lakes maps and it is obvious that the GLH is extremely map dependent and situational. I have been playing around with several of the LHC maps and I think now that I overestimated the GLH in isolation. It can be powerful even then but in isolation with no offshore islands it is not all that great. And as discussed elswhere, if one has two or three medium-sized islands one may run into the annoying colonial expenses which are only mitigated by the additional trade. (I really think that this is a bad feature making these maps even harder than they already are.)

Very map dependent indeed. It's quite the gamble to dedicate so much resource to an early wonder in a MP game that will see the settling of max 3-4 cities before borders will start encroaching.