So long as the adamantium suit isn't treated like Superman's skin I see no issue with that.

I am not particularly set on a Modern Setting.

I am hesitant to set it in any particular period or year rather than a nebulous "near future" for the same reason I am hesitant to set it in the United States or any real country, or during the known tenure of any President.  

If folks would prefer to set it in another time period that would be fine, but one of the advantages of setting it in a country that is essentially a fictionalized Cuba (with elements of Brazil) is that it becomes relatively easy to justify have the dictatorship vibes you described and also to have something like the cars and technology of another era if people are more interested in that.

Cuba is known, among other things, for it's classic American cars as a result of embargoes from the US.

A more violent Spiderman would be very interesting to see, but you might also take that the other direction.  A Spiderman in a very violent world, with police he can trust less, with a government he can trust less, without an Avengers or an X-Men to help him when and if things get heated.  Less hospitals to take wounded civilians too if he webslings into a burning building to save a family of four.

An interesting arch (if you are doing an Origin or a Brand New Hero sort of story) might be her starting out with a No Killing Rule, and quickly realizing the very real danger she is in despite her powers.  Struggling to hold herself to that standard despite the increased restraint it requires and the increased danger that represents.

But then I'd also get a kick out of Spidey just webbing some asshole up and swinging him/her hard into a wall several times.

This reminds me of a story I really liked, probably influenced this idea, details in the hider below

[hider=My Hider]
I believe it was set in either the Warhammer Universe or a like Days of Future Past XMen Universe.
The premise was basically a gang in a vaguely post-apocalyptic world where the police had completely abandoned the streets.  Gangers shoot it out pretty openly, control food supply, and vie with each other constantly for control.  The story was basically about a small gang as one of them sought to take control from the current leader.  This was all during a small war that quickly got out of hand. 

 Young runner comes back telling them that "Oh My God, they killed her" and they're coming here.  Big shoot out ensues.  Lots of cool action.  Lots of the gang, who you've come to like well enough, get absolutely wasted.  Lots of the attacking gangers get obliterated by Gang Leader Dude.  At one point there are only like 3-4 of the gang left alive when they find out that their leader has powers that basically make him invulnerable to the other gangers guns.

To me at that point it becomes quite interesting.  How different things become when the leader of your gang knows the other gang simply can't kill him.  You die just as easy as anyone else, but the guy telling you to protect him or to go hit the other guys has no reason to fear being killed.
[/hider]