I've never read any of Joy Harjo's works before, so it was really interesting reading this one! I ended up getting a bit carried away, so I enspoilered my longer answers and gave a quick summary under them.
1. Spoiler
The main feeling I felt from the poem was a sort of warning or alarm. I also felt like the poem has some bitterness, especially towards the middle, which resolves into hope near the end.
At the start of the poem, the language used feels very matter of fact. The narrator is stating what they're trying to do ("make a map for // those who would climb through the hole in the sky"), how they'll do it (using the "desires of humans"), and how to use it (not to be "read by ordinary light").
Then the poem becomes very dark and unhappy, with descriptions such as fog that "steals our children while we sleep", "flowers of rage", "monsters...of nuclear anger", and "trees of ashes". The narrator goes from perhaps an almost neutral tone, to a sombre, warning tone.
Then, after probably scaring the listener with these warnings, the narrator tries to supply some map or action plan for moving forward into the future. There isn't any forced positivity, and the emotion feels very raw and honest, yet just the implication that there /is/ a future seems to suggest some reason for hope. It feels like there's some resignation or acceptance for the situation. The narrator concedes that "an imperfect map will have to do", and that "we were never perfect", but also says that "fresh courage glimmers from planets".
The poem ends with an almost devastating, but also an inspiring statement: "you must make your own map". Nobody wants to have to stumble through the dark without a guide; they want instructions. But it also leaves so much room for hope and improvement, and empowers the reader with the idea that they have the ability to change the future.
In short: I felt like the poem had a sort of warning tone, and maybe even some slight bitterness. Towards the end, however, it becomes more hopeful. The poem begins in a matter of fact way, simply stating information for the reader - what the narrator wants to do, and how they will do it. As the poem progresses, it grows a lot more negative, using darker imagery such as "steals our children", "anger", "rage", and "trees of ashes". But the narrator gains hope at the end, saying that "fresh courage glimmers from planets". It ends on a very moving line: "you must make your own map". This is perhaps a scary concept but it also leaves the reader with a sense of hope.
2. Spoiler
At first I thought the poem was about death and coming to terms with it. Partly because of the title - I thought "the Next World" meant whatever comes after death. And I took "climbing through the hole in the sky" to mean leaving the physical world behind for outer space/heaven.
However, when I reached the lines saying to "take note of the proliferation of supermarkets and malls, the // altars of money. They best describe the detour from grace", I realized that this might be more about the modern lifestyle, especially consumerism and our relationship with nature/the environment. Phrases such as "trees of ashes", "tracks of the monster slayers", and "cities of artificial light", seem to imply that at the present we have a negative relationship with the environment. This is compared to how in the past we (the author's ancestors) once knew "the names of the birds", and "everything in this lush promise".
In addition, I think there is some criticism of the general attitude of society - "Flowers of rage spring up in the depression" seems fairly literal; "Monsters are born there of nuclear anger" is a bit less straight forward - I personally interpret this line to be talking about nuclear weapons, and how so much power has the potential to make anyone into a monster.
As a whole, I think the poem is criticizing modern society and advocating for a change in ways, or a new "map" - however it isn't prescribing an answer, it's basically saying that we need to figure out our mess, for ourselves.
In short: I thought at first that the poem was about death, especially because of the title (Next World -> afterlife). However when the poem began describing supermarkets and malls, I realized it might be more about the modern culture and some of its shortcomings, especially consumerism, and our relationship with nature. The narrator contrasts the present to the past: in the present, we have "cities of artifical lights", whereas in the past, we knew "the names of the birds", and "everything in this lush promise" (now forgotten). As a whole, I think the poem is saying that we have to change some stuff about how we're currently living (follow a new map).
3. Spoiler
The imagery doesn't fall into one particular family, and feels fairly disconnected from one stanza/line to the next. For me, this adds a certain sense of confusion and ambiguity, which fits really well with the style and subject of the poem. There does seem to be a strong emphasis on forgetfulness, though, as she makes a reference to this in several places and through different imagery.
In the legend are instructions on the language of the land, how it
was we forgot to acknowledge the gift, as if we were not in it or of it.
Keep track of the errors of our forgetfulness; the fog steals our
children while we sleep.
We no longer know the names of the birds here, how to speak to
them by their personal names.
What I am telling you is real and is printed in a warning on the
map. Our forgetfulness stalks us, walks the earth behind us, leav-
ing a trail of paper diapers, needles, and wasted blood.
And then, near the end of the poem, a shift:
Remember the hole of shame marking the act of abandoning our
tribal grounds.
This use of language emphasizes the contrast between the past and the present, and creates a sense of nostalgia on the narrator's part. I also think using this repetition of the words "forget" and "forgetfulness" gives a sort of connection between all the disjointed imagery.
In short: The imagery is pretty disconnected, which I feel complements the subject really well. A sense of continuity was created though, through repeating the words "forgetting" and "forgetfulness". This focus on forgetfulness also emphasized how things have changed from the past and creates a feeling of nostalgia. Towards the end of the poem she switches to "remember", which changes the mood subtly and ends on a more hopeful, future-oriented note.
4. I think that anyone can understand and appreciate the poem, because she uses very expressive language and imagery that can stand on its own. However, there are instances where an understanding of Native American legends would add to your experience of reading the poem. As
@Liminality mentioned, the "white deer" probably references a legend, and understanding what it was referring to would add to your understanding of that line. But overall, I think you can definitely still appreciate and enjoy the poem regardless of your background.
-Liminality you mentioned structure, which I thought was really interesting! At first glance the poem seems to be lacking much structure and feels pretty loose and almost rambling, but I did notice that in several places she used line breaks really deliberately.
Trees of ashes wave good-bye to good-bye and the map appears to
disappear.
^In this pair of lines, which could very well have just been one line, she decides to break it up into two lines, the second of which is only one word. I think it could be to highlight the contradicting nature of words "appears" and "disappear", but that's just one possible reason!
What I am telling you is real and is printed in a warning on the
map. Our forgetfulness stalks us, walks the earth behind us, leav-
ing a trail of paper diapers, needles, and wasted blood.
^In this section, I think it's really interesting that she divided up "leaving" between two lines. I'm not entirely sure why, I'd love to hear if anyone else has any ideas!
Whew! I think I'm out of things to say

I honestly had way too much fun doing this - I know Liminality already said this, but thank you so much
@alliyah for organizing this!!