1 - How does it looks like in general (that is, outside of a host)? An empty grid, with black sky and hovering hosts above?
A friend told me it was more like a "first-person game" rendition of wherever I logged in, but the book's description looked much more abstract.
So what it looks like varies a lot. So your character life style dictates what grid they log into, but there a great deal of description on this subject, however the canon description are, all major corp grid are beautiful designed and use BTL tech to make you feel happy to there, all them also are themed according to the megacorp they are made from, Ares American Utopia etc. As with through out SR history indvidual hosts should be themed to match your image of them, and can be anything. From Dynasty warriors, Underwater ship wrecks, to chapter from pride and prejudice, all those have been discussed in cannon somewhere.
2 - Virtual Positioning
I dont quite get the idea of "locations" in the matrix. Sure, in a host it's shared VR with every other person there - I get that. But if I jump into VR while commuting in the subway, or at a friend's, where do I show up in cyberspace? A generic entry point? A virtual rendering of where my meatbody is? Something else?
So you describe it as the character logging into the regional grid (RTG) or a corporate grid. What that looks like you should work out before, but most are discussed in a source books. Now they are just moving around, then they are just using AR (Augmented reality), and then they are just looking at the world around them with the addition of bunch of new cooler signs, and art.
3 - Matrix Reach
Can I reach ANYTHING connected to the Matrix (aside from noise/grid penalties) as long as I can see it (or browse it)?
Is it possible to have a "secret club" in the matrix, only visible to those invited?
(also, sorry if this is not the place to make these questions - let me know if thats the case and sorry for the inconvenience).
You don't need to see it. You can find anything in the connected matrix, Full Stop. If you can see it, you don't need to find it, you know where it is. You can't make a secret club, you can defend things via slaving them to something with serious firewall. But the downside of you being able to find anything is they can find anything that connected as well.
4 - Matrix and runs.
Scene: a shootout is going on;
A street samurai and his decker buddy are taking cover behind their ford americar.
Across the street, an enemy ganger is behind the dumpster, with a smartlinked gun.
The Sam and the ganger exchange shots while the decker jumps in VR to look for a way to screw up the ganger.
Rules aside, how can that be described once he plugs in?
How does he know which icon is the ganger's gun/cybers/PAN so that he can mark it and do his actions?
And how does it look like from his perspective?
So there multipe ways to look at this and deal with this. In the combat situation if your building a decker to brick stuff, your best method isn't full VR, AR is better, so long as the decker has some decent meat would initiative. AR is the smarter way to do it b/c it's preferable to have distraction penalty then to be limp body in gun fight. There is a lot to be consider in this. But I think you're miss what AR is totally. Bricking (Breaking a piece of tech by maxing out it's matrix damage CM) is very doable in 5e, based upon how 5e is setup, it's very likely guns will be wirelessly active. (Why? b/c smartgun need to be wireless active to get their full stats, further, if someone isn't wirelessly active they are going to draw attention from the cop.) The examples are discussed in the Core, and expanded on in data trails. There is a of combat options that lets a decker attempt to brick something before initiative is even rolled.
How you do it systemwise is dataspike, you just build to be really good at attack, you don't mark anything, you just throw data spike, and brick right through whatever firewall they have it slaved too. It's very possible to this, but it require building to do it, and it take decent amount of optimization.
Now if you're actually looking at matrix runs, you VR up, go into login grid, (Ether via the RTG to corp grid), or you go to the location of the Host. Keep in mind, grids, buildings (Hosts), can be infinitly big, but it's simply a question of finding the thing you want to get into and then connecting to it. That process is discussed in the core.
5 - Game Depictions of Matrix
Any of the shadowrun videogames (SNES/Genesis/"Returns") do any justice to the depiction of the matrix?
The video games are good. But consider them to be an RTG, they have budget constraints that will keep them from hitting the real scope, of what the matrix is in SR. VR is all the senses. Sight, smell, touch, hearing, and yes even taste. You feel the sun, smell the better the life air, hearing the beautiful themed music, etc etc.