So, the first requested question is this: What's your opinion on whether there will be animals on the "New Earth?"
Let me begin with a big, important answer.
I don't know.
Seriously - this is the most important part of the (perhaps) long, drawn out answer to follow. I don't know. In fact, I would say that I cannot *know* - because this is not revealed in the Scriptures. What follows is not knowledge based upon Scripture; it is not authoritative like the proclamation - it is a derived speculation.
So, having said that I do not know, I am left to make my best guess.
My best guess is yes.
Why is this? Well, we do get the visions in Scripture where it does speak to the Lion and the Lamb laying down together... so there is animal imagery. However, that could very well just be imagery of things being idealized and at peace and safe (you have the child sticking her hand into the viper pit... but we will not be given in marriage, so I don't know if there will be "children" in heaven... I tend to speculate that we will be adults... I can't say that conclusively... I don't know precisely what is entailed in John's idea that we will be like He is, but Christ is an adult... so I lean that way). So I can't point conclusively to that.
I also note that we have the idea of a new heavens and a new earth. I would assume that these will be like unto the present heavens and earth, just perfected. As the present heaven and earth have animals... I'm guessing so. Of course, the Revelation of John says that there is no need for Sun or Moon as New Jerusalem gives its own light... but again, that's vision language (is it speaking to physical light, or to the fact that all order will be directly apparent from God - that we will not need the Sun and Moon to order days and seasons as we do now...)
In other words - it really beats the tar out of me.
If there are animals - it will be good.
If there are no animals - it will be good.
I do tend to lean towards there being animals, simply because they are part of the physical world - and the New Creation will be a physical place. Part of man's creation was to tend to creation - the animals are the things we were given the task of naming. I'd guess that part of my completion, my perfection in the life of the world to come will be a fulfilling of this stewardship role in toto.
So, I think it makes sense to say that there will be animals. But then, the new Creation is what eye has not seen and what ear has not heard -- so it may be totally beyond what I think or expect... yet it will be good.
I don't expect my old dog Ditto to be there. I don't even know if there will be weiner dogs there, or even dogs at all, or even canines (although if Joshua had Caleb, and Jesus is the same name as Joshua, and Caleb is the Hebrew word for "dog"... oh, wait, that is stretching it). I know this won't bother me, for every tear shall be wiped from my eye and that the former things will be remembered no more. But who knows - God is creative and restorative... maybe there will be Ditto there... and if there is, he won't pee on the trash can any more.
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I think it can be fun to speculate about heaven, as long as we remember a few things.
1. It does mostly end up being speculation. Expect to be wrong.
2. Don't speculate yourself into despair. It will be good.
I also remember being in second grade and reading Revelation. Then I had a dream, one of those dreams that seems to last a long, long time, much longer than a simply night's sleep, and I dreamt (normal dream, not prophetic) that I was in heaven. And it was neat. Then I woke up and was disappointed that I wasn't there.
That may be the most faithful I've even been concerning heaven. I don't know. But again - He shall come again, and it will be good. The specific shape and form of that good is something that I don't need to know.