Your attempts look pretty good.
Among the few questionable parts, though, the most important would be your verb choice of 「住んでいる」. Native speakers' choice would definitely be 「生きている」. To live in a certain age/era is 「生きる」. This mistake is actually kind of common among J-learners because in English, both are "to live".
The other "mistakes" are fairly minor compared with that verb choice.
"Among 重要, 大切, 大事 which is the most relevant to mean "important people"?"
Most definitely, 「重要」. It is just about the only choice out of the three.
One cannot, however, say 「重要ように」; One must say「重要なように」 or 「重要であるかのように」.
"Has という been used correctly here?"
Yes, it has (and I was impressed by it).
"In my Japanese translation above, it seems there are 2 adverb of time which is 時代 and 今. However, the English translation, 今 is modified by 時代. Therefore, how to make ~時代 the modifier of 今? How about the following?"
Easy. Use 「今の時代に」、「この時代に」 or 「今この時代に」. All are almost equally good. You could also use 「[現代]{げんだい}に」 instead.
Your phrase 「スマートホンの時代である今」 is actually correct and even natural by the native standard. Other choices would include:
「このスマホの時代に」、「今、このスマホの時代に」、「スマホ[全盛]{ぜんせい}の現代に」, etc. (I myself like the last one, but you are the author.)
Other minor points I want to mention:
「忙しくて」 sounds conversational. I would drop the 「て」 if I were you. To make it sound even more formal or academic, you might consider using 「[多忙]{たぼう}で」.
「見える」 can be left as is, but 「見えてしまう」 and 「見えがちである」 might be a little better word choices.
Finally, I would suggest that you use one more comma as the sentence is fairly long.
An example that uses as much as what you used would be:
「無教育な怠け者が、まるで多忙で重要であるかのように見えてしまうという、スマートフォン全盛の現代に我々は生きている。」
One cannot use 「今」 together with 「現代」, so I dropped it.
