The Talmud in Sukkah 28b states the following:
ת"ר כל שבעת הימים אדם עושה סוכתו קבע וביתו עראי כיצד היו לו כלים נאים מעלן לסוכה מצעות נאות מעלן לסוכה אוכל ושותה ומטייל בסוכה מה"מ דת"ר תשבו כעין תדורו מכאן אמרו כל שבעת הימים עושה אדם סוכתו קבע וביתו עראי כיצד היו לו כלים נאים מעלן לסוכה מצעות נאות מעלן לסוכה אוכל ושות' ומטייל בסוכה ומשנן בסוכה איני והאמר רבא מקרא ומתנא במטללתא ותנוי בר ממטללתא ל"ק הא במגרס הא בעיוני
Our Rabbis have taught, All the seven days, one should make the Sukkah, his permanent abode and his house his temporary abode. In what manner? If he had beautiful vessels, he should bring them up into the Sukkah, beautiful divans, he should bring them up into the Sukkah; he should eat and drink and pass his leisure in the Sukkah.
Whence do we know this? — From what our Rabbis have taught: Ye shall dwell implies, in the same manner as you ordinarily live. Hence they said, All the seven days one should make his Sukkah his permanent abode, and his house his temporary abode. In what manner? If he has beautiful vessels, he should bring them up into the Sukkah, beautiful divans, he should bring them up into the Sukkah,’ he should eat and drink and pass his leisure in the Sukkah; he should also engage in profound study in the Sukkah. But it is not so? For did not Raba say, Scripture and Mishnah [should be studied] in the Sukkah, but Gemara outside the Sukkah? — There is no difficulty, The former [statement refers to] revising, the latter to profound study.
(Soncino translation)
It sounds like anything that can be practically done in the Sukkah should be done in the Sukkah.
In any case, the custom of taking the lulav in the sukkah is attributed to R. Isaac Luria in Sha'ar Hakavanot:
והנה טוב הוא שאחר עלות השחר שהוא כבר יום קודם שתתפלל תטול הלולב בתוך הסוכה ותברך עליו ואה"כ הד' נענועים של הלל יהיו בבית הכנסת ומה טוב היה אם היית מתפלל בסוכה כדי שתתפלל ותטול הלולב בברכתו בשעה הראויה אליו שהוא אחר תפלת שחרית קודם ההלל
And it would be good if after dawn when it is already day, prior to praying, you should take the lulav inside the sukkah and make the blessing on it, and afterwards the four wavings of Hallel should be in the synagogue. And how good it would be if you would pray in the sukkah, in order to pray and take the lulav with its blessing at its appropriate time which is after the morning prayer before Hallel.