是的，最近在忙我的finals,英语却越来越退步了。

Finer translation details

1.   I had chosen to translate directly because "rainbow" does not sound too poetic and is too direct here.  It is always good to let the reader use their own imagination.  It is the same in Chinese not to use 彩虹 or even 霓虹.

2.  As you have pointed out 三仙島 and 三島仙 are completely different things.  The first one is the immortal isle where as the latter one is the the immortal from such a place.

3.  步盡 from the context means to wander all over the place, complete going everywhere and not "the end of the wandering".

1.  The Three blessed isles of the Eastern Ocean in Chinese mythology where immortals live.  They are 蓬萊, 瀛汌 and 方丈.

2. Literally "separated on two banks".  Here "bank" does not mean the rive bank but "the shores separated by a vast ocean". Benben, this footnote is provided because the translator is making an assumption on the word "bank" and that translation is technically not exact.

3. See 1

Finally, you have grasped the meaning of the last line but may not have the target language fully in command to express it well.  This is what I mean that for any translator the command of the target language must be much better than the source.  A professional translator on the other hand must have the languages fully in hand.

見而不踏七彩橋, Seen but not stepped upon, is the seven colored bridge of above,

聞而不去三島仙. Heard but never been to the Blessed Isles 1 to see the immortals.

難鎖詩魂兩岸隔, Difficult it is to contain poets' souls on their shores 2

步盡蓬萊一遊客. For their imagination can wander on the Immortal Isles 3 as a guest. 

However, this is not the case with your translation for the first two lines.  "solving problems" and "heard in poems" are purely your conjectures and interpretation which in nowhere has the poem ever hinted.  Sometimes a poem is crouched in such ambiguous terms, the translation should reflect the same ambiguity than to 2nd guess the poet.  This is a trap that most inexperienced translators fall easily into.

In the last line, for the two most difficult characters you opted to do a transliteration.  This is perfectly acceptable as there is no direct equivalent in the target language.  However, you need to provide footnotes.  This is because most readers would not know what it is.  It does not matter how common it is the source language.  Remember, the translation is intended for the target audience and not everyone of them is steeped in the culture of the source language. Footnotes must be provided whenever there is no accurate translation.