Is it ever permitted to daven out of order?

Ideally, a person should not daven before hanetz hachamah. If one is unable to wait until that time, he should wait until misheyakir. From misheyakir, he may put on his tallis and tefillin and say Shema along with its brochos. If, however, a person must travel at alos hashachar and will not be stopping until after zman tefilah, Shulchon Aruch rules that he should say Shemoneh Esrei before he leaves and wait until the appropriate time to say Shema and its brochos. Although he will not be connecting geulah to tefilah, the ability to daven Shemoneh Esrei while standing trumps the benefit of davening in order. Others disagree and hold that he should wait until the proper time and daven in order, even if he will have to sit during Shemoneh Esrei. The custom follows this opinion.

סימן פ"ט ס"ק ל"ח-מ"ב

Why is it forbidden to daven while standing on a high place?

When davening Shemoneh Esrei, it is forbidden to stand on any sort of platform (e.g. a chair or pillow). There are three reasons for this prohibition: 1) A person who is not standing on firm ground is concerned that he may fall. These thoughts will disturb his concentration. 2) It is haughty to daven to Hakodosh Boruch Hu from a high place. 3) Tefila is akin to the service in the Beis Hamikdosh, where the Kohein would stand directly on the floor.

סימן צ ס"ק א' ביאורים ומוספים דרשו 2

Where should the chazon stand in shul?

Davening on raised ground which is three or more tefachim high is forbidden. Although there is no concern that one may fall, it is still considered to be haughty. If the raised platform has the dimensions of four square amos or is surrounded by partitions, it is classified as a domain of its own and is not considered to be raised in comparison with the rest of the room. There is a custom for the chazon to daven from the lowest place in the shul. A source for this minhag is found in the pasuk “From the depths I called to You, Hashem.”

ס"ב ס"ק ה' ו' ז'

• According to the Zohar, as of midnight, a person who was sleeping may not eat upon arising. Nonetheless, it is recorded that one of the great masters of Kabbalah ate and drank upon arising (before alos) when he was weak. The Achronim explain that the Zohar only forbids eating for pleasure, not eating to sustain the body for avodas Hashem.

• One of the activities which is forbidden before davening is Talmud Torah. There is a concern that one may become so engrossed in his learning that he will forget to daven.

• Contemporary poskim debate whether showering is permitted before davening. Some are lenient if one needs to wash himself before going to the mikvah. Others are lenient if one showers without soap. Some say that if it is difficult to daven without showering, it is permissible to take a shower. 

• Is it permissible to daven beneath the open sky?

• How many windows should there be in a shul?

• Is it permissible to daven behind a shul?