Background
The first thing to be aware of is that this poem was composed in Chinese by the poet 于 濆 (Yú Fén) in roughly 874. (Brief Chinese Wikipedia article about the poet here.) As such, the Japanese version must be viewed as a translation. And if you've ever done much translation yourself, particularly of poetry, you've probably come to understand that certain liberties are often taken in order to produce a target text that 1) is meaningful, and 2) works at least somewhat as poetry.
Here's the original as best I can tell, with the modern Mandarin readings and a rough English translation.
花開蝶滿枝，花謝蝶還稀。
Huā kāi dié mǎn zhī, huā xiè dié hái xī.
Flowers open and butterflies fill the branches, flowers say goodbye and butterflies go back to being scarce.
惟有舊巢燕，主人貧亦歸。
Wéi yǒu jiù cháo yàn, zhǔrén pín yì guī.
However, there are old-nest swallows, the house owner is poor and yet they come back.
Specifics
With that in mind, let's address your questions.
- Why is
還 read また? I found the kanji in Jisho, which says that it can be used instead of 帰 in 帰{か}える, but I could not find it being used as また. Is this some kind of play on words based on this kanji's meaning of returning?
As Chinese, 還 has various senses, of which the "returning, going back" one seems most relevant here. In the context of the poem, the text is describing how the butterflies go back to being rare, which can be expressed in Japanese using the word また.
- Why are
還 and 亦 both used for また? Is there a negative connotation in the former, and a positive in the latter? What about 又?
Again, the Japanese for this poem is a translation. The use of the Japanese word また fits the context, the grammar, the meaning, and (to a lesser extent) the meter.
The kanji are, strictly speaking, hanzi (漢字 as read in Mandarin).
- In the first case with 還, the text is describing the butterfly situation returning to a previous state of few butterflies.
- In the second case with 亦, and using the although sense for the Chinese term, the text is describing that the swallows mentioned in the previous line return home even though the master of the house is poor. In the Japanese, this although, even though sense is conveyed using the particle も, and the また was probably added to reinforce the 歸【かえる】 meaning and to fill out the otherwise-missing syllables needed to approach the expected meter.
- What is the meaning of
有あり?
As noted in the comment by Sweeper, that's a typo. It should be 有【あ】りて, which you correctly guessed is an archaic form of modern あって.
Digression: alternative approaches to translingual poetry
As 漢文訓読【かんぶんくんどく】 (literally "Chinese text, [Japanese] meaning-reading"), one of the common renderings appears to be:
花【はな】開【ひら】けば 蝶【ちょう】枝【えだ】に滿【み】つ
花【はな】謝【しゃ】すれば 蝶【ちょう】還【また】稀【まれ】なり
惟【ただ】 舊巣【きゅうそう】の 燕【つばめ】有【あ】りて
主人【しゅじん】 ��【まず】しきも 亦【また】歸【かえ】る
Going slightly further and reworking a bit to better fit the Japanese poetic meter preference for lines of 5 and 7 morae, we might get:
花【はな】が開【ひら】けば
蝶【ちょう】枝【えだ】に満【み】ち
花【はな】が謝【しゃ】すれば
蝶【ちょう】また稀【まれ】に
惟【ただ】舊巣【きゅうそう】の
燕【つばめ】が有【あ】りて
主【ぬし】貧【まず】しきも
また歸【かえ】る
And treating this more as a translation than just 漢文訓読, we might even change some wording around while maintaining the meaning, particulary reworking the end.
花【はな】が開【ひら】けば
蝶【ちょう】枝【えだ】に満【み】ち
花【はな】が謝【しゃ】すれば
蝶【ちょう】また稀【まれ】に
それはともあれ
古巣【ふるす】の燕【つばめ】
家【いえ】主【ぬし】が
貧【まず】しく然【さ】るに
また帰【かえ】る
Please comment if the above does not fully address your questions.
