A simple answer to your final question would be to simply rotate the Sefer Torah on the Bimah by 90 degrees counterclockwise (where the face is the surface that the Torah is resting on), and then unroll and lift it from the right side of the Bima (where a Gabbai usually stands), directly showing those on the right, and then turning around.
Another possibility is that "showing" a particular side of the room could mean holding it open for them to look at, rather than quickly moving it through their sightlines.
In regards to the entire order of the Shulchan Aruch (which is based on Maseches Sofrim 14:14), אנציקלופדיה תלמודית כרך ח, הגבהה ב notes 2 approaches to Hagbah, as discussed in the Sheyarei Kenesses Hagedolah (OC 134:7, Hagahos Beis Yosef). One opinion is that the Magbiah should simply lift and turn around with the Sefer Torah, thus not taking the specific order of the Shulchan Aruch literally. The other opinion (if I understand it correctly) is that he essentially lifts and flips the Sefer Torah around to show the side with the writing on it to those 4 sides of the room, in order. This sort of solves the issue because he is simply flipping the side of the Sefer Torah with writing on it, as opposed to turning around. (Presumably the Magbiah would do this by originally pulling the left wooden support of the sefer up in the air, and only once that had been lifted would he lift the right side upright. Following this, he would be required to awkwardly twist his arms in order to show the Sefer Torah to the other sides of the room.) This sounds very complex, and at least as problematic as opening and closing it in the air.
Here is the text of the אנציקלופדיה תלמודית, mainly quoting the Sheyarei Kenesses Hagedolah:
כשמראה פני הכתב לעם יש סוברים שהמגביה עצמו נשאר עומד במקומו, אלא שהוא מסובב ומחזיר את הס"ת כדי שפני הכתב יהיו לעם העומדים בצד ימינו, ואחר כך הופכו ומראה פני הכתב לעם שבצד שמאלו, ומחזיר לפניו והופך אחורי הכתב נגד פניו כדי שהעומדים לצד מזרח יראו הכתב, ואחר כך הוא מחזיר הכתב נגד פניו שיראו העומדים לצד מערב, ויש סוברים שהמגביה עצמו הוא סובב לכל הצדדים, וכן המנהג.
See also Rav Menashe Klein (Mishneh Halachos 11:103), who asks a related question and discusses this at length.