This article lists several laws related to both the internal placement of items as well as design of the synagogue building, itself. I am citing only those items related to the building design.
Architecture: The Noda Biyhuda (tinyana, Orach Chaim 18) writes that there is no formal obligation to build the shul with four walls in correspondence with the design of the Temple. He permits building a shul in any shape, provided that the intention is for reasons of space and convenience, and not to emulate non-Jewish architecture.
Doors and Hallways
The interior design of a shul is halachically prescribed, and also relies heavily on comparison to the Temple.
Tosefta in Megillah 3:14, teaches that the doors to a shul must be constructed on the eastern side. Just as the doors of the Temple opened westwards, so too, the doors of a shul should open to the west. In the Temple, this design ensured that upon entering, one would bow before the inner chamber. In a shul, one must do so in the direction of the Aron Ha-Kodesh – the ark containing the Torah scrolls. From this Tosefta, the Shulchan Aruch (150:5) derives that doors to a shul must be positioned so, that one who enters is facing the direction of the Aron. (The Aron’s location is determined based on the direction of prayer in the specific area.)
An interesting addition to this is found in the rulings of the Chasam Sofer.
The Bach (Orach Chaim 90) learns from the Yerushalmi that a hallway must be constructed as an entranceway to a shul. This halachah is also quoted by Magen Avraham (35) and Mishnah Berurah (61). Based on comparison with the Temple entrances, the Chasam Sofer (Orach Chaim 27) rules that the doors to the hallway should not be aligned with the doors to the shul itself, but should rather be to the north and south (assuming that the doors of the shul are in the east).
In reality, it seems that not many shul architects are familiar with this halacha.
Trees: Rabbi Akiva Eiger (glosses to Shulchan Aruch 150:1) quotes Rav D. Arama who prohibits planting trees in the courtyard of a shul. This corresponds with the prohibition of planting trees “adjacent with the altar of Hashem.” Authorities discuss this prohibition (see Piskei Teshuvos 150:19, note 90), which applies specifically to trees (and not to bushes or flowers). Some adopt a more stringent position and some a lenient stance.
My comment re the direction of the ark & doorways - The upstairs of my shul has the ark facing south. The downstairs chapel's ark faces east. I don't know why the upstairs was designed that way.
In contrast, if you have visited the Touro Synagogue in Newport Rhode Island, the shul building faces diagonally compared with the surrounding street grid. I understand that when built, the shul building was designed according to the above specs that the ark face east and the doorway is opposite. I assume that the street grids were designed a century or later after the shul was built.