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
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
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.