Again, if I can talk out my rear a bit, I think it's because the 'burning' temperature of the cage material is checked, but the fact that nether cap stays at freezing temperature irreguardless keeps it from being harmed. Though, interesting the cage isn't burning or anything.... you might want to check that.
In terms of attracting migrants, you have the following conditions to consider (most of which you can check on the 'z' screen):
1. How much wealth is in my fort? The more wealth you have, the more popular you are.
2. How much wealth have I traded to people? How much of it have I offered? The most that goes out, the more migrants that come in.
3. How many caravans or officials have successfully left without dying to !!fun!!? Every caravan or official that leaves spreads information about your fortress. This is part of the reason why your migrant wave in your second winter (after the dwarven caravan in the autumn, one year after you arrive) is so gigantic compared to the others.
4. The first couple of migrant waves are 'hard coded' - that is, until about year two, you're going to get migrants every season. I don't know if Dwarf Fortress just makes them out of thin air or what, but even if your chosen Dwarven civilization is near-dead, you'll still get migrants.
5. The problem for many people - especially as they start getting into years 4-6 - is that there's TOO MANY MIGRANTS, and you start running into logistical issues providing for them all. That is, at least, under normal settings; many people hard-cap their population to prevent this. Personally, I deal with it by making a 'temporary' base in the dirt layers, then once I have my feet under me, start constructing and erecting an awesome fortress beneath me, with like planning and forethought and supplies and slave labor and stuff.
Also, don't forget that the more wealth you have, the more goblin (and other) attention you're going to draw. While this isn't as sure in this edition as it was back in, say, 2012, as enemies of your civ can decide to attack other cities in your civilization instead of you, this means that if you make expensive, masterwork steel armor valued in the hundreds of thousands of dwarf bucks for your military, you're probably going to need that armor sometime in the next few years.
IMO, the best way to quickly value up your fortress is to make an awesome textile industry, get yourself some decent gem cutters and setters, and start pumping out clothes. Clothier and leatherworker level up quickly, and a pig tail / above surface weavable plant farm, along with necessary dyes, is relatively easy to set up. You can also breach your first cavern for silk, though that is (of course) a riskier proposition. Once set up, you can stamp out masterwork clothes, sew images onto them (despite the name, the material for this is cloth, not thread), the cloth going INTO those clothes will be masterwork (weaver also levels up quickly), and you can also encrust them to produce multi-thousand dwarf buck clothing left and right. Don't forget about prepared food, either; as your farmer's grower skill levels up, he'll be able to plant large stacks of seeds in single tiles, resulting in larger stacks of plants, which cooks can mash together with other stuff to produce dwarven prepared food barrels worth upwards of 10K (which you can also trade under the 'barrels' section. Note that you'll want rock pots to trade to those persnickety elves). This, among many other reasons, is why you want to select 1-3 dwarves and specalize them in nothing else but planting stuff; your whole farm and soft good industry will get more efficient as a result.