These answers are of course based on my (possibly wrong) understanding so take it with a grain of salt.
1. Emptiness is a quality of things. There is no emptiness without objects. (HHDL's Heart Sutra commentary)
2. Emptiness is a translation. Other translations include voidness. Don't analyze the word. Things are not hollow or lacking in qualities. Emptiness simply means that things are empty of inherent existence. (Emptiness by Tashi Tsering) Not empty of anything else.
4. Everything exists but not inherently. Not by itself but from conditions. Now you may say "Well duh, trees comes from seeds" but that's a mistake I made. It's deeper than that. Without branches, leaf, space etc. there is no tree. If it existed by itself it still would. Yet, the tree does exist due to conditions. Thus things lack inherent existence. Google Hume bundle theory for a good explanation of this.
5. Everything exists, just not inherently.
6. If things have conditions to exist they do not exist inherently. If things exist inherently they cannot have conditions to exist. These conditions bringing a thing into existence is dependent origination.
If something dependently originates it cannot exist on its own right and is thus empty. Also, because things are empty they must have dependently originated.
7. Sure why not.
Nature of mind is another discussion, it is empty (of course just like every thing else) but in this context it refers to things like Clear Light.
It gets a little more in-depth than just seeing that "things" are only dependent on constituent qualities (such as a tree is dependent on branches, leaves, space etc..). There's different "tiers" or levels of the emptiness investigation and it's application to reality. In seeing that nothing inherently
exists separate from causes and conditions the study actually has to descend to the most fundamental of levels in order to have a profound effect, otherwise it merely stays on the level of conceptualization(which is all well and good, but there's "deeper" realizations to be had).
This "emptiness" investigation is seeing that nothing exists separately from causes and conditions. So it's true on a conventional level that a tree isn't separate from it's branches, leaves, space etc... but to leave it at that; you're still left with branches, leaves, space etc.. (for the sake of what i'm trying to describe i'll treat 'space' as a thing) and these are 'things' as well. The evaluation can continue further and further to deconstruct branches and leaves down to their constituent particles(and the particle down to their constituents) but this process never ends and really never leaves the realm of the intellect. All of this evaluation is going on within the mind, fundamentally using concepts which are no different than the 'tree' you began with.
From here it gets closer to what was said above about separating the 'experience' of emptiness from the 'conceptualization' of it.
So what's been said so far is still using emptiness on the tier or level of deconstructing 'things' to constituents which ultimately end up being other 'things'. But what does this deconstructing or conceptualizing depend on? Depends on the mind. And when one looks at the mind, you see that the mind is made of constituents called thought and memory. So so far this intellectual deconstructing is dependent on mind(aka thought) and the mind(aka thought) is dependent on that which the thought is conceptualizing. So there is no mind separate from thought/memory... and no thought/memory separate from that which they(thought/memory) objectify. What is objectified is not separate from the thought/memory... and the thought/memory is not separate from mind. If that can be assimilated thoroughly what's seen is that there's no separation between any of them. And that there are no 'things' (branches, leaves, space) separate from conceptualization. And the collection of conceptualizations (in time; which is dependent on conceptualization) constitutes that which we call mind.
Once 'things' such as branches, leaves, space are seen to dependently exist on concepts (and the concepts dependent on those 'things'). And the concepts are seen to be dependent on mind (and the mind a conglomerate of concepts) one starts to see that a web of dependent origination starts to form and that these different designations are only a product of conventional language. Apart from the conventional language(which is useful!) these 'things' do not inherently exist.
Now at this point one will say "ok but i still see 'things', i still experience these things"... even if they're divorced from the notion of inherent existence on the level of conceptualization, the sensual 'happening' of life and reality is still there. I'm still here seeing sights, hearing sounds etc... so this can't be left at this level.
About this Nagarjuna states: "All of this is but one's mind," That which was stated by the Able One, is to alleviate the fear of the childish; it is not [a statement] of [final] truth.
At this point the investigation reaches yet another tier or level and this is where it really starts to have a profound effect. The sensual experience has to be investigated. Take the experience of that which we'd label a branch for example; in the investigation of the 'branch' you'll notice that this 'appearance' isn't separate from the sensory perceptions with which it's experienced. The 'branch' is dependent on the sensual experience of it. So the 'branch' is predominantly 'seen' and 'felt'(in addition to the other senses). You can even say the 'branch' is composed of the 'seeing' of it and the 'feeling' of it. Apart from the visual and tactile sensations, there is no 'thing'(branch). So the 'thing'(branch) is dependent on the sensory experience. But for this to be a true application of emptiness both sides of this have to be accounted for. And that comes like this; without the 'sight'(branch) there is no 'seeing'(sensory perception) of it. So the appearance of a sensory perception is dependent on it's percepts(objects of perception). So with this being ascertained; why even conceptualize two different designations(perception, percept)? Both designations are mutually interdependent co-arisen imputations. An 'object' is the cognizing of the 'object' and the 'cognizing' is dependent on the appearance called an 'object'. They are not two different things.
And this goes for every sensory perception.
From here however... the same investigation is directed onto the subject and applied to the position of the 'cognizer' a.k.a. the 'self which is perceiving'. One sees that the appearance of the self(subject which perceives) is dependent on the perceiving... and the perceiving is dependent on the perception. So no piece of the trifecta stands alone or has any inherent existence. They all collapse all the way down and the existence of the triad(perceiver, perceiving, perceived) is merely a conventional imputation overlaid onto a seamless 'happening'.
About this Nagarjuna states: The cognizer perceives the cognizable; without the cognizable there is no cognition; Therefore why do you not admit that neither object or subject exists at all?
From here the next tier or level goes to the awareness, or consciousness, or 'knowing' of this.
About this Nagarjuna states: The mind is but a mere name; Apart from this name it exists as nothing; So view consciousness as a mere name; Name too has no intrinsic nature.
The consciousness/awareness is also of dependent origination and therefore empty.
Time is also of dependent origination and therefore empty.
Emptiness is of dependent origination and is therefore empty.
These investigations allow for one to account for and see the dependent origination of every 'thing'.
And that these 'things' being of dependent origin lack inherent existence.
So if apperceived directly beyond mere conceptualization the reification of these 'things' as being concrete ceases. That includes yourself.... and the knowledge of this. So even though a realization is had, the realization likewise is empty.
I would've broke this down further but i gotta go!