I’ve just signed up to this forum basically to read some user opinions, as I’m about to purchase a 500 Twin Air. So, I can’t comment on the 500, but I’ve done lots of mods to my cars over the years, which have all been ‘city car’ to ‘supermini’ size. Some of the things I tried years back were indeed a waste of time - usually the ‘more power’ ones. So I don't do as much these days.
But moderate lowering springs up to 30mm I would recommend for spirited drivers. That is about 10% of the spring length. It’s not about springs being ‘hard’. They are progressive and have as much give as normal springs at first, only becoming ‘harder’ further into the suspensions travel. It’s mainly about lowering the centre of gravity: with that being closer to the axle height, then the moment of the centrifugal force is reduced, so the tendency for the body roll is less. This regardless of damping.
It has not made any of my cars crashy or left them bouncing all over the road. Far from it. It has improved body control when cornering, stability, feel and susceptibility to cross winds. I have never grounded one in 20 years.
Some manufacturers endorse the option of lowering for keener drivers. For example, Toyota’s own tuning arm (TTE) offer their own suspension lowering kits. Ford sell Eibach lowering springs as official Ford accessories! Skoda have lowered some Fabia’s simply for the improvement in economy.
RE someone’s bad experience lowering a Suzuki Swift Sport. I’m not altogether surprised. A reputable company like Eibach’s main offerings are usually 30mm, but less for the warm and hot hatches as their springs are usually already lowered (by the manufacturer - it’s one of the first things they do to improve handling!) Typically a hot hatch will be 10-15mm lower than standard cars. To fit 40mm lowering springs to an already lowered Sport version is rather extreme.
On the main thread title, yes insurance will cost you more. Some companies won’t touch a modified vehicle, others not those with increase power, some have limits to how much you can do - 1 mod, 2 mods or 3 mods, after which they won’t offer insurance. The latter approach seems sensible as a large number of vehicles have in fact been modified in some way - non-standard alloys, tow bars, or a dealer fit accessory. They don’t want to lose those customers.
I’m currently with Aviva - they’ve been good. It put my premium up, and by more than a couple of insurance groups worth, but not extortionately.
All of that would be fine for pretty much any car on the market. The 500 comes standard with pretty woeful suspension. It needs decent dampers first, not springs.