I am going to answer them in reverse order just because #3 is easiest to address and #2 is not so bad either.
Most importantly, protection from lightning. Trees, by their nature,
are tall organisms and the taller an object is, the more attractive it
is to a bolt of lightning.
The lightning is going to strike regardless of whether the village is there. I do not recall any of the structures providing any lightning rods above the treetops, and I just Googled some images to confirm. The village should not make lightning strikes worse.
If you just mean "The lightning that would already have struck is now closer to you." then yes, perhaps that could be an issue. That would be similar to all tree-houses that humans actually have built; I know of nobody who has had a problem with their tree houses being too close to lightning. Of course, they probably are not in use during a lightning storm, but then, maybe the residents of Lothlorien generally avoid the upper levels when lightning is striking close.
For a parallel, look to modern humans who live in dangerous areas. After hurricane Katrina, lots of people said New Orleans should be abandoned. Instead, people moved back in and rebuilt even though that was, in my opinion, worse than what close lightning strikes would do to a treetop village. So even if this were a problem, it can be set aside by persistent, stubborn peoples.
Constructing bridges from hundreds of feet off the ground. (Vines I
will not allow because they might work for carrying Ewoks from tree to
tree, but not heavy goods.)
To get the bridges there, they can do it the same way people used to make bridges across chasms. If it is close enough, throw your rope. If it is too far, you need to tie off at one end with a rope long enough to go down, across, and back up, then pull it up.
As far as the material of the bridge itself, I challenge the assumption that vines would be insufficient. One or two vines, maybe, but building a bridge made up of many vines would work, though I still would not go that route.
Instead, there are plenty of other materials that can be used, and have been used in reality. Just think of the famous bridges that people used to make out of grass which have inspired the look of many bridges on the big screen.
Making ropes is something that has been done for thousands of years, including ropes which can hold hundreds or thousands of pounds. In fact, I have practiced rope-making with natural resources, so I can tell you that I can go outside and make a rope out of grass that can hold my weight, and I have done so. I have never tried to make very strong ones, but all that requires is weaving more strands in to make a bigger rope, and the strength goes up exponentially with the diameter, so I have no doubt that I could go outside my house and, using what I find lying around naturally, make a rope that could carry me and heavy goods. The techniques required are simpler than they sound, but it does require a lot of time and patience.
Once you have enough rope made, you can connect parallel ropes either with more ropes, or you can lay wood across them.
Building structures strong enough to hold but light enough to not make
the trees top-heavy.
I think this is the hardest one to answer definitively and might require an engineer in that field to be certain. I answer it with comparisons instead of numbers.
I will point out that, if you use large trees - imagine the redwood and sequoias that this type of village would probably be made on in reality - the weight of the village would probably be a fraction of the weight of the tree trunk it is attached to. Especially if you are careful in your construction; you can use relatively thin and lightweight wood. So it is doubtful that it would tip over.
If your top-heavy concern is for the weight to crush the bottom of the tree, I would like to remind you that some large trees have very large holes cut completely through them, including the famous sequoia tree that you can literally drive through (Google images: "tree with road through it"). In this case, if the severely weakened trunk can support the immeasurably massive weight of the tree above it, then I doubt the added weight of a village would crush a tree that does not have such a huge hole through its bottom.
To reinforce this point, think about wood building supports. A 2x6 can hold a lot of weight even when it is horizontal, even if you do not support it in the middle. Multiple of them holding up a floor are going to support practically whatever you want. Some types of wood are very strong. If a thin length of wood run horizontally can hold so much, an entire vertical tree is going to support a lot, lot more.
You can look up wood weight-load data that will tell you how to figure out how much weight a given type and thickness of wood in a certain orientation will support. It does have its limits, but if lumber can support entire large buildings, and you generally don't even consider anything beyond 6 or 8 inches thickness in wood for private housing (and that's for the load-bearing parts that support everything else), then surely a tree which is that many feet across will support a mind-bogglingly lot of weight.
If you are concerned about the top-heaviness in the wind (ie: blowing over), I will point out that trees already have a lot of surface area in their leaves that catches a lot of wind. In fact, if you stay inside a dense forest during a windy time you will notice that the wind is reduced inside the forest, though I do not recommend this since trees already fall down in high winds even without buildings in them.
If you are worried about wind, to reduce the amount of wind caught by the tree structures, I would suggest that a lot of it be as open as you are willing to make it. Reduce the amount of walls and roofs. A lot of it can be just floor to walk around on, and since floors are horizontal they will catch less wind, though still enough to get torn off if you don't build them well. Use flat or very shallowly sloped roofs in areas with no walls. Try to make your fully-enclosed (wall & roof) areas be small and hug closer to the tree trunk - that is, a small circle centered around the tree trunk with a diameter just enough for cramped sleeping and a few other minor activities.
If you have a large storage area, make it large vertically instead of horizontally and have it encircle the tree and use a spiral staircase around it, all this preferably lower down on the tree, in fact, all your biggest structures should be lower on the tree, and smaller ones should be higher up. This might not even be necessary; your people could do some tests by trying to build the worst, most top-heavy, wind-catching buildings high up in trees (these are tests; they won't be lived in) and see how they react to time and storms, then use that as building guidelines.