How does one determine the authority behind "proper" biblical interpretation?

I won't be able to answer your question fully (because it is, as you have acknowledged, a vast topic), but here is a start.

One of the stock arguments of those who reject the authority of God's Word
is:  "That's just your interpretation.  People just interpret the Scripture any way that suits them."  This reminds me of Bill Clinton's (in)famous
statement:  "That all depends upon what your definition of the word "is"
is."  In time, everybody came to know about his sexual relationship with Monica Lewinski.  It wasn't really a question of the meaning of the word "is".  I think that whenever we depart from the plain and simple meaning of Scripture, we are headed for trouble.  I think that whenever we seek to set aside a command of Scripture, we are headed for trouble.  John 7:17 is a very important verse to me:

"If anyone wants to do God's will, he will know about my teaching, whether it is from God or whether I speak from my own authority" (John 7:17, NET Bible).

When Jesus gave the Great Commission He instructed the disciples (and thus, the church) to make disciples by "teaching them to observe all that He has commanded" (Matthew 28:20).

Those who really want to obey God's commands can expect to have a clear sense of what God is commanding us to do.  Those who are willing and eager to obey our Lord do not seem to have so much difficulty determining just what it is that He commands us to do.

Those who do not wish to obey will inevitably come up with some creative way of avoiding the command.  Who we are determines who we heed (obey):

    One who acts wickedly pays attention to evil counsel;         a
liar listens to a malicious tongue (Proverbs 17:4, NET).

I think a couple of texts of Scripture are particularly important to our
conversation:

8 "And when he comes, he will prove the world wrong concerning sin and righteousness and judgment - 9 concerning sin, because they do not believe in me; 10 concerning righteousness, because I am going to the Father and you will see me no longer; 11 and concerning judgment, because the ruler of this world has been condemned. 12 "I have many more things to say to you, but you cannot bear them now. 13 But when he, the Spirit of truth, comes, he will guide you into all truth. For he will not speak on his own authority, but will speak whatever he hears, and will tell you what is to come. 14 He will glorify me, because he will receive from me what is mine and will tell it to you. 15 Everything that the Father has is mine; that is why I said the Spirit will receive from me what is mine and will tell it to you" (John 16:8-15).

14 The unbeliever does not receive the things of the Spirit of God, for they are foolishness to him. And he cannot understand them, because they are spiritually discerned. 15 The one who is spiritual discerns all things, yet he himself is understood by no one. 16 For who has known the mind of the Lord, so as to advise him? But we have the mind of Christ (1 Corinthians 2:14-16).

These texts inform me that the unbeliever is incapable of grasping the full meaning of God's Word.  The Holy Spirit was sent not only to inspire the writing of the Scriptures (2 Peter 1:21; see also 2 Timothy 3:16), but to open the spiritual eyes of those who are blind, so that they will be convinced and convicted regarding what God has said (John 8:14-16).  And as believers it is the Spirit of God who continues to convey the meaning of Scripture to believers (1 Corinthians 2).

When people tell me, "That's just your interpretation," I try to respond in these ways:

First, I would try to stay with the words of Scripture themselves, rather than my own paraphrase or interpretation of them -- "The Bible says 'There is no one righteous, not even one' (Romans 3:10).  Jesus said, 'I am the way, and the truth, and the life.  No one comes to the Father except through me' (John 14:6).  Etc.  These plain statements are hard to take in a non-literal, non-plan way.  Folks may deny them, but it is because their meaning is too clear; not because their meaning is obscure. Sometimes it may be worthwhile to ask, "If the Scriptures were absolutely clear to you, and their meaning certain, would you obey them?"

Second, I might say something like this:  "The Bible teaches that God's Spirit convicts us of the truth of His Word.  While I can only convey God's Word to you, the Holy Spirit is the instrument that works within your heart to convince you that God's Word is true.  And so my suggestion is this:  Try to forget the Scriptures that we have read.  But if the Spirit of God will not let you forget them, and if He convinces you that they are true, then I would suggest you take them seriously and act on these truths accordingly."

Third, pray.  The apostles devoted themselves to "prayer and the ministry of the Word" (Acts 6:4).  We must "preach the Word" (2 Timothy 4:2), but it is God who will make that Word come to life in the hearts of men, transforming them.  Thus, we must proclaim the Word and pray that God will not allow His Word to return void:

10 The rain and snow fall from the sky and do not return, but instead water the earth and make it produce and yield crops, and provide seed for the planter and food for those who must eat. 11 In the same way, the promise that I make does not return to me, having accomplished nothing. No, it is realized as I desire and is fulfilled as I intend" (Isaiah 55:10-11).

12 For the word of God is living and active and sharper than any double-edged sword, piercing even to the point of dividing soul from spirit, and joints from marrow; it is able to judge the desires and thoughts of the heart. 13 And no creature is hidden from God, but everything is naked and exposed to the eyes of him to whom we must render an account (Hebrews 4:12-13).

It is God's Word that is inspired and authoritative.  Proclaim that Word and trust in God to work in the hearts of men.

Now I should not close without saying one more thing.  We need to be careful to distinguish between

    Those things we know to be true (what the Bible     dogmatically
and authoritatively claims as truth)

    Those things which we hold as our personal conviction.

    And, those things which are speculative and uncertain.

Sometimes we are dogmatic about things which are not fundamental, about things which devout Christians differ over.  Let us stick to the fundamentals, and proclaim them authoritatively.