I feel clueless about some basic things. Please enlighten me.
Can Sigil's portals lead to every layer of a plane, or just to it's first one?
How's the City of Doors actually connected to the Outland? Is there a permanent portal to somewhere, or do they appear somewhere in the Cage and lead to random destinations in the Outlands? (at first I thought the Gatehouse would be that thing, but that's as wrong as possible)
Teleport, gate and similar spells only bring you to a plane's first layer, right?
Is it possible to travel the planes solely by foot, without using planar paths and gates?? For example, could a cutter start her journey in the Outlands, march past Curst until reaching Carceri and then struggle her way to Hades, instead of using the Curst Gate and travelling on the Styx?
If there are differences you know of between all the mechanics in 2nd and 3rd edition, please tell them.
Sigil's portals can lead anywhere. Absolutely anywhere.
There are some permanent portals leading to various gate-towns in the Outlands, and there are lots of temporary or one-way portals leading to various parts of the Outlands as well. The City of Doors is in the Outlands, hovering an infinite height above the Spire in the center of the Land. At least, people assume it is, though it's hard to confirm that. You can't see the Outlands from Sigil; if you look over the city's edge, you just see a whole lot of nothingness. Perhaps this is because the Spire balances out all things until only nothingness remains, or perhaps there's another reason Sigil is hovering in the middle of a seemingly endless void. Some sources say that if you jump over Sigil's edge, you'll end up falling forever toward the Outlands, never reaching it, but another source (the original Planescape boxed set) says you end up in a completely random plane.
Gate spells can reach any layer - it's a 9th level spell, after all. However, some planar lords and gods will block access to their realms. Plane shift can reach whatever layer you know the proper tuning fork to. The first layers are reached using single notes, while deeper layers require combinations of notes. Usually the second layer is a sharp, the third layer is a flat, and deeper layers require exotic combinations of tones and materials. Teleport Without Error can get you to any places that have been studied first, though the spell doesn't permit planar travel in 3rd edition. Astral Spell gets you to the Astral Plane, where color pools only reach the first layer of each plane in 2nd edition; this isn't true in 3rd edition.
No, this isn't possible. If you pass Curst you end up in the Hinterlands, an unexplored, misty region of the Outlands more subjective and strange than the core part between the gate-towns. The Hinterlands are infinite; you can travel them forever and never reach anywhere. If you turn back, though, the gate-towns are always only a few days away, no matter how far you've traveled.
Teleportation didn't require access to the Astral Plane in 2nd edition. In 3rd edition, the Astral Plane touches all layers of all planes unless specifically noted otherwise, while in 2nd edition it only touched the first layers of the Outer Planes and the Material Plane.
In 3rd edition, Teleport spells don't permit interplanar travel, while in 2e they do.
The 3e Manual of the Planes says it's possible to walk between adjoining Outer Planes (though not between the Outlands and any other Outer Plane - the Hinterlands still exist in 3e) without a portal, there being "thin places" in the infinite planes where the two planes join. In 2nd edition, the only way to cross between Outer Planes was through magic or a portal.