For further understanding let me dissect the capture you shared, this can improve understanding.
How do we interpret LSU in OSPF?

- A OSPF packet header which is part of the packet fields of an LS Update. Contains: version#, type, packet lenght, router id, area id, checksum, autype, authentication.
B Link State packet (Type 4), out of the 5 OSPF packet types.
C The LS Update packet carries besides the OSPF header, two things: # of advertisements, and the Link-state advertisements.
- D This is the common LSA header. Router LSA (Type 1) consist of it plus what comes next.
- E In this case the LSA 1 fields start of with options (V, E and B bits) and the number of links.
Four different Link Type's exist for LSA 1:
- Point-to-point (Type 1)
- Transit (Type 2)
- Stub (Type 3)
- Virtual Link (Type 4)
- F1.1 Stub network
- F1.2 Point-to-point to another router
In regards to:
This is a kind of a very basic question in OSPF packet exchange. What are these kinds of link in LSU - Stub link, PTP and again Stub link.
The Link type is one of the fields used in the Router LSA, all 4 were mentioned previously. The information in the router LSA's link ID and link data fields is associated with the type of link OSPF is operating.

Following with:
What I am trying to achieve here is to understand how a particular node in the simple topology knows how to install routes in the routing table.
Like, how does it know what next hop it should take for a particular
destination. I understand that it will run Shortest path first
alogrithm to determine that.
For this there is a short answer concerning 3DB's: OSPF loop avoiding
Basically each node put's itself at the root of the tree and determines the lowest cost and loop free path for each destination.
Concerning:
So, I am looking into LSU to understand that they are advertising.
For example, in this diagram, it is advertising - 4.4.4.4 Data
12.12.12.1 Metric 10.
These are broad questions? 1) What does data mean here? 2) Does it
mean that in order to reach 4.4.4.4 my hop is 12.12.12.1. 3)How the
other router knows how to reach 12.12.12.1 because that router isn't
connected with 12.12.12.1 interface?
The reason why does IP's fit the way they do:
"An OSPF router always forms an adjacency over a point-to-point link using an unnumbered interface. This causes the Link ID field to contain the router ID of the adjacent peer. the Link Data filed contains the IP address of the local router's interface or the local interface index value for unnumbered interfaces." JNCIS study guide by Joseph M. Soricelli
4.4.4.4
is an OSPF Router ID, not an IP address. Both are 32-bit numbers and often use the same dotted-decimal notation, but they are not the same thing. You could have an OSPF router with the4.4.4.4
Router ID, but also have the4.4.4.4
IP address somewhere else on the network.