God's foreknowledge according to the Calvinist?

As a former Calvinist, I hope I can help provide more light than confusion. To caveats: (1) at the outset you should know that there is diversity even within 'Calvinism' so several answers are possible on each point, and (2) I'm not entirely certain I understand you questions but I will do my best to help on each of the four points you note.

1. In brief, a Calvinist believes neither of these statements.

1.A. GotQuestions is correct when they say Christians cannot infer from the bible that there is such thing as a pre-existence of souls. Because of this, Calvinists do not believe that God had a loving relationship with pre-existent souls. Verses that GotQuestions cites bear this out (e.g. Zech. 12:1) and this is the standard Calvinist position.

1.B.I. Of course 1.A. depends logically on 1.B. in assuming that to 'fore-know' means to 'fore-love'. The reference offered by Ligonier ministries shows that the concept of 'knowing' throughout the bible is often (but not always) a relational term. This is true. However, this does not rule out cognitive knowing as one potential meaning of 'knowing'. E.G. in one sense I cognitively know information about you from your question, the sources you cite, and use of language. In an another sense, in as much as you are a real person who wrote this question, I know you relation-ally - but only insofar as I cognitive-ly understand your question. Because of this, Calvinists do not have to say, and to my knowledge most do not say, that God had a loving-relationship with pre-existent souls. Because, per 1.A. they do not believe in pre-existent souls and per 1.B. knowledge of persons and their future within your mind (i.e. cognitively previous to their existence/creation) fulfills the basic meaning of foreknowing.

1.B.II. I hypothesize that when Ligonier Ministries tries to say that God 'fore-loved' they don't intend that connotation to be used in reference to predestination but in contrast to the Arminian view which bases God's cognitive "knowledge" on the choices of individuals. Therefore, they want to avoid a type of foreknowledge that is cognitive and potentially based on the choices of free human persons, and they opt to argue that knowledge is relational not cognitive. (On their part this is a false dichotomy but that is another issue entirely)

Conclusion of 1. You may be correct in pointing out a contradiction between what Calvinists believe and what they logically should believe, but they do not believe either of these things. Alternatively, you could be using Calvinist definitions of words like 'fore-knowledge' in an overly mathematical way which is confusing you (1=A, A=/=B, a=B, therefore A=B AND a=/=B). Calvinists do not believe in a pre-existent soul nor in a pre-creation relationship of God to the 'elect'. Except in the case of Jesus. Jesus is both pre-existent and eternally loved by the Father through the Spirit. The corporate Arminian explanation from this point disagrees with Calvinists by saying God has eternally elected only Jesus and through his actual fore-loved-ness he elects others.

2. I don't think you understand what Ligonier Ministries is trying to say in their post.

2.1. (1) The pot analogy is an analogy is an analogy designed to say one thing not a description of reality that should be pressed beyond its original purpose. (2) In the Calvinist view, it is a reference to God's free determination of certain groups and/or persons (i.e. Israel in Rom. 11) to be saved from their sins or punished for their sins. This strong use of the word fore-knowledge (+determination/predestination) is what Calvinists mean and in their post, Ligonier Ministries says this is the correct view as opposed to the weaker view of fore-knowledge held by some Arminians in which God simply fore-knows (-determination) what will happen.

Conclusion of 2. Because Calvinists believe in a stronger, deterministic, type of fore-knowledge (i.e. writing, directing, visualizing, and then watching a movie) which also means predestination, they believe Arminians view of simple-foreknowledge (i.e. just watching a movie someone else wrote) is incorrect because is DOES NOT leave God as Sovereign. You seem to be quoting what Calvinists DO NOT believe, but this seems to be more the fault of the writer who puts this where a conclusion usually is. In reality Arminians do not necessarily believe this, rather that God writes, directs, visualizes, watches and himself acts/participates in a story where he has created real actors besides himself.

3. Per discussion under question 2.1. you are correct in believing this is NOT consistent with the sovereignty view of Calvinists. They believe God isn't forced to love pre-existent souls, and they believe, like the potter, he is free to make people things for good or bad purposes. However, some moderate Calvinists will allow along with Arminians that God can sovereign-ly choose to create a world where he includes the free choice of creatures (at minimum Adam and Eve had free will) in his designation.

Sources: IVP's Divine Foreknowledge: Four Views would give you far more context about the overall perspectives possible, but I hope this has been helpful.

Conclusion: At the back of this is the deterministic view of Calvinism where God exists beyond time, and actually created time itself. At the end of the day, your question revolves around the paradox of a a-temporal God relating to temporal (time-bound) creatures. I'd consider exploring William Lane Craig's Time and Eternity for a better answer on this count, although it is a very dense debate in the philosophy of religion.