Why is it permissible to hold a lulav during davening?

During Pesukei D’zimrah, Kriyas Shema or Shemoneh Esrei, one may not hold an object which would cause him distress if it fell to the ground. The focus on holding the item would disturb his concentration on davening. Examples of objects which are forbidden to hold include tefillin, sifrei kodesh, knives, money, a loaf of bread, keys and walking sticks. It is also forbidden to attend to a child while davening.  One may hold a mitzvah object if in doing so he is fulfilling a mitzvah. Therefore, a person may hold his lulav during davening on Sukkos. In case he starts worrying about dropping the lulav, the affection for the mitzvah will aid in refocusing his kavanah. Similarly, one may hold a siddur while davening.

סצ"ו ס"ק א' ב' ג' ביאורים ומוספים דרשו 4

May one walk to a bookshelf to check something in a sefer during Shemoneh Esrei?

If a sefer falls in the middle of Shemoneh Esrei and the fallen sefer is disturbing one’s kavanah, picking up the sefer is permissible at the end of a brocha.  It is forbidden to motion or point to get others to pick up the sefer. If one becomes confused about the text or needs to clarify a time-sensitive halacha while in the middle of davening, he should motion for someone to bring him a siddur or sefer. If this is impractical, he may walk over to get the sefer himself. Afterwards, he should remain in his new place for the duration of Shemoneh Esrei unless something there disturbs his kavanah. 

ס"א ס"ק ז' ביאורים ומוספים דרשו 10, 11, 12

Preparing the siddur before davening

A person who wants to daven from a siddur should make sure that the siddur is available before he starts Shemoneh Esrei. If he will be saying any additional sections, these passages should be prepared before davening; he should not be leafing through the pages during Shemoneh Esrei. There are special benefits to davening from a siddur. The Achronim say that it is also appropriate to look inside the siddur during Chazoras Hashatz. Doing so will serve as an aid to listening to the chazon.

ס"א ס"ק ח' וט'

• Ideally, one should avoid davening in a place containing idols. Precedent for this practice is found in the account of Moshe Rabeinu’s refusal to daven in the idolatrous cities of Mitzrayim.

• Before starting Shemoneh Esrei, one should clear his throat and nose, straighten out his hair, and remove anything which may distract him.

• One should face downward and close his eyes while davening. A person who does not shut his eyes—except for one who is looking at his siddur--will not merit to see the Shechinah when he passes away. 

• Is yawning permitted during Shemoneh Esrei?

• Is spitting permitted during Shemoneh Esrei?

• May a tallis be adjusted during Shemoneh Esrei?