Q:

HTTP request from client to server. The TCP/IP stack journey

I am trying to understand the process a HTTP request does from host to server in terms of packets and hardware it passes through.
First of all, the HTTP data is put into the a TCP packet (layer 4). This one is then packed into IP packets (layer 3), framed by the NIC (layer 2) and transmitted (layer 1) to the closest router over the physical medium (cable, fiber, etc).
Question 1. Does the NIC (the one connected to the client's LAN, since the router has two NICs) in the router reconstructs the frames and passes the payload (so the IP package) to the router's processor? It seems logical, but I have not been able to find anywhere online where this is explicitly said.
The router then has an IP package and forwards it according to its routing table through the other NIC, which frames it again and transmits it. 
Hop-by-hop, the NIC of the server finally receives & reconstructs the frames and passes them to the ...(See Question 2)..., which unpacks the TCP package and somehow the HTTP information is read and the operation is executed with the arguments given.
Question 2: Kernel? Is it the one responsible for IP packaging/unpackaging inside the computer?
Question 3: How is responsible for TCP and "understanding" HTTP in the computer? What part of the OS is it? I am guessing this one is also the responsible for understanding HTTP.
I am aware there are many more details, but the overall question would be too long. I am also happy to see interesting links to deepen into the matter.

A:

Question 1. Does the NIC (the one connected to the client's LAN, since the router has two NICs) in the router reconstructs the frames and passes the payload (so the IP package) to the router's processor? It seems logical, but I have not been able to find anywhere online where this is explicitly said.

I don't know what you mean by 'reconstruct' here. The NIC receives the frame and passes it to the CPU, probably via DMA.

The router then has an IP package and forwards it according to its routing table through the other NIC, which frames it again and transmits it.

The word is 'packet', not 'package'. Otherwise correct.

Hop-by-hop, the NIC of the server finally receives & reconstructs the frames

Again I don't know what 'reconstruct' means. The frames are received.

and passes them to the ...(See Question 2)..., which unpacks the TCP package and somehow the HTTP information is read and the operation is executed with the arguments given.
Question 2: Kernel? Is it the one responsible for IP packaging/unpackaging inside the computer?

To the IP layer of the networking stack, which passes it up to the TCP layer, which eventually passes the data to the server application.

Question 3: How is responsible for TCP and "understanding" HTTP in the computer?

The TCP layer is responsible for understanding TCP. HTTP is an application layer protocol and is understood by the application.

What part of the OS is it? I am guessing this one is also the responsible for understanding HTTP.

There is no 'it'. There are two separate questions here with two separate answers: see above.

