What exactly is Infinite Nature?

This is a difficult question because "Infinite Nature" is not a term from scripture per se, and as a theological term it leaves much to be desired.  The term infinite is a difficult term philosophically because it means "something without any bounds or attributes", because something without any bounds or attributes cannot be described, that thing would be difficult to distinguish from nothing.  For example, imagine a triangle without any sides — basically a triangle without sides is nothing.  Some philosophers call this a "bad infinite", and it is important that Christians avoid ascribing this bad infinite to God because he has revealed attributes, and, in a good sense, these attributes limit Him.  In other words, he is good, therefore not bad; he is wise, therefore not foolish; etc.  Because of the "bad infinite", some theologians prefer to use a term such as "perfect" instead of "infinite" to describe God and His attributes.  Thus, we would say God is "perfectly loving" instead of "infinitely loving", etc.  If you were to choose to use the term infinite to describe God, the best way to use it would be to make the point that God is not subject to or under the power of any created or finite thing.  However, even here, it might be more useful to use "Lord of" all creation, time, space, etc.  

So how does this notion of God's perfection and Lordship have to do with time and space?  Well in relation to time, we often say that God is eternal.  Different theologians have taken this to mean different things.  For several centuries in the church, it was held that divine eternality meant that God was atemporal, or outside of time. A group in the church called the Socinians, however, denied that eternality meant atemporality, insisting that it merely meant that God had no beginning or end.  In line with the Socinians, recent theologians have denied that God is outside time, and because he is inside time, they insist, He cannot know the future, since it hasn't happened yet. These developments have made it necessary to further define what we mean when we say that God is eternal, and once again it is helpful to use the language of Lordship to better encapsulate what the scriptures say about God's relationship to time.  You will remember that I said that the best possible way to use the word infinite in relation to God would be to make the point that God is not subject to created things.  In much the same way, it is important to make the same point in referring to God's eternality.  Specifically related to time, God is not subject to the following limitations: (these are taken from theologian John Frame's Doctrine of God)

1.      The limitation of beginning and end — God has no beginning in time, and he will have no end. (John 1:1 "In the beginning was the wordà")

2.      The limitation of change — God is not subject to time in that it changes him in ways beyond his control. (Malachi 3:6 "For I am the LORD, I change not; therefore ye sons of Jacob are not consumed.")

3.      The limitation of ignorance — God is fully aware of the past, the present, and the future.  (Isaiah 46:9-10 "Remember the former things of old: for I am God, and there is none else; I am God, and there is none like me,  Declaring the end from the beginning, and from ancient times the things that are not yet done, saying, My counsel shall stand, and I will do all my pleasureà")

4.      The limitation of temporal frustration ¡— Everything happens exactly according to God's timing. (2Peter 3:8 "But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day.")

Because God is not subject to these limitations of time, and because he is all-powerful, God is Lord over time.  This is what we should be emphasizing when we point to God's eternality.

We see then that God is Lord of time.  Is he also Lord of space?  I think that, based on scripture, we can safely answer yes.  To paraphrase your question, you asked whether God's infinity in regard to space was the same as his immanence. The answer depends once again on whether you mean a bad infinite, or a good infinite.  In a bad infinite in regard to space, once again, God would have no boundaries that set him apart from space, and thus would be indistinguishable from the space.  If you mean, however, that space cannot set limitations on God, so that God, while distinguishable from his Creation, is free to do as he pleases with it and in it, than yes, one aspect God's infinitude in regard to space can be called his immanence.  Immanence in theology refers to God's "closeness" to creation, his ability to meaningfully interact with what he has created.  This is contrasted with his transcendence, or his ability to "rise above" creation and maintain a creators eye view distinct from creation itself.  Immanence, however, is probably not the best word to use to convey the sense that God is "infinitely big".  The term that many theologians have chosen to convey this idea is "immensity".  Immensity is intended to convey the biblical idea that "the heavens, even the highest heaven, cannot contain youà" (1Kings 8:27).  God is so immense that he is not limited by the physical world.