There's not really a hierarchy. It's really more of a web. There are national (or international) ISPs that connect to a large number of other ISPs. There are smaller ISPs that connect to only a few other ISPs. Then there are large companies that have their own backbone (Google, Microsoft, Amazon, Facebook, etc).
There's no "rule" about who can ...
Don't bend fiber like that, it's sensitive.
The black patch cable might be damaged, replace.
Check the SFP modules - for gigabit over MMF you'll need 1000BASE-SX (850 nm).
Verify SFP compatibility with the converters - quite a few vendors force you to use their branded modules.
You can check a fiber port with a simplex cable as loopback.
One converter shows ...
Cisco is being a bit literal. 10/100 "T" (twisted pair) does have independent TX and RX conductors. In a point-to-point situation, the link is fundamentally full-duplex. However, when a hub is involved, everyone's TX is connected to everyone else's RX. As a result, it's impossible for more than one node to transmit at a time, thus half-duplex, but ...
You'd need to look at the frames in detail.
While simple (=unmanaged) switches only forward incoming frames, managed switches can support a multitude of features and protocols that may cause them to send frames on their own, e.g.
spanning tree BPDUs (or for shortest path bridging 802.1aq)
LACP control frames for link aggregation
GVRP, MVRP for dynamic VLAN ...
No, those are routers, not switches. You can use subinterfactes on a router interface to have layer-3 interfaces into separate networks (VLANs) as a trunk to an external switch:
encapsulation dot1Q 10 ! Tag traffic for VLAN 10
ip address 10.0.10.0 255.255.255.0 ! ...
(This will tread on "historical trivia")
The tiered construct was the original vision of the ARPANet (what became the internet.) The modern internet does not work like that. In fact, the early internet -- and ARPANet before it -- didn't entirely work like that either. With the advent of BGP, any network could be anywhere, connected to anyone. ...
It seams you are need to Configure InterVLAN Routing - I believe it can be done with 3560 you have. Try ip routing command - and if possible go to https://www.cisco.com/c/en/us/support/docs/lan-switching/inter-vlan-routing/41860-howto-L3-intervlanrouting.html to get full instruction