What is the Ashkenazi minhag as regards wearing a tallis gadol during Kabbalas Shabbos (i.e. the ceremony/liturgy of welcoming in Shabbos on Friday evening)? Should it be worn, or not?
The contemporary normative Ashkenazi approach to the topic generally follows the BaH (OH 18:1):
בתענית ציבור כשלובשין טלית במנחה כשמגיעים לברכו של ערבית יסיר הטלית
מעליו דכיון שאין לובשין טלית אלא לקיים מצות ציצית נראה כאילו סובר לילה
זמן ציצית הוא אם לא יסירנו מעליו מלבד ערב יום הכפורים שלובשין מבעוד
יום לאחר מנחה ואין מסירין מעליהם בלילה לפי שדומין למלאכים לובשין לבנים
ומתעטפים לבנים
With the exception of Yom Kippur, a talleth is not worn at night as doing so erroneously signals that one has reasoned that the night is a zeman sisith (a period within which one fulfills the miswah of sisith). This position is endorsed and cited authoritatively by the Magen Abhraham (OH 18:1:1), and the Mishnah Berurah (OH 18:1:6).
The Shaliah Sibbur however wears a talleth during all prayers, even at night, as a function of כבוד הציבור (the honor due to the congregation). See MB ad loc.
is there a kabbalistic reason why some customarily wear a tallis gadol during kabbalas shabbos?
First it is worth reiterating that there is firm halakhic basis for this practice. As you note this was the conduct of the Amora R. Hanina (Shabboth 119a). One also finds earlier evidence of this practice in the Tannatic era as well, Shabboth 25b reports this as the conduct of R. Yehudah b. 'Ilai:
כך היה מנהגו של רבי יהודה בר אלעאי ערב שבת מביאים לו עריבה מלאה חמין
ורוחץ פניו ידיו ורגליו ומתעטף ויושב בסדינין המצוייצין ודומה למלאך ה'
צבאות
Further one might argue against the BaH's concern, that wearing it on Friday night does not signal that one has reasoned that it is a zeman sisith as it is apparent that one is wearing it in order to greet the Shabboth (just as he maintains that it is apparent on Yom Kippur when an alternative consideration is at play).
In terms of surveying the practices extant in the larger Jewish world: Temanim, who historically largely accepted the Mishneh Torah as authoritative (as you note H. Shabboth 30:2), have preserved this practice (I have observed this at several Baladi minyanim). However there are other pockets of Jewry that retain the practice as well, such as some Hasidic sects (Karlin, Slonim, Lelov and Toldoth Aharon) that don it before Minhah and and take it off either after Maaribh or Qiddush. Even among Hasidic sects that do not generally observe this practice, many of their rebbes/admorim do. Anecdotally, I have also been told that some Moroccan families continue this tradition as well.
Now, to return to your question. On the more esoteric side of things it's a bit of a mixed bag. The Radbaz wrote (Mesudath Dowid, miswah no. 84):
ואין מצות ציצית בלילה משום לא תחסום שור בדישו דא"כ קשרת מדת הדין שלא תפעול בלילה ואסור לבטל מנהגו של עולם
Accordingly, R. Haim Vital reported that R. Isaac Luria (who was a student of the Radbaz) was generally careful to remove his talleth before the onset of sunset (Sha'ar Hakawanoth, Tefilah ha-Minhah, Derush 2). Of this camp, R. Immanuel Hai Ricci and R. Yosef Gikitila both castigated the practice of wearing a talleth at night, lest one arouse דינים קשים (difficult judgements).
The Yalqut ha-Gershoni (OH 18) accordingly questions, on what basis do the various rebbes who wear one on Friday night elect to conduct themselves contrary to the aforementioned kabbalistic considerations:
ועיין בספרי קבלה המתעטף באותו שעה גורם כמה ענינים ומעורר דינים קשים...עכשיו בימינו נתפשט המנהג בין הרבנים והצדיקים להתעטף בערבי שבתות וימים טובים בתפילת ערבית ולא ידע על מה סמכו
Some argue that the kabbalistic sources that speak against wearing a talleth at night do not intend to include Friday night (see Birkhath Eliyahu, p. 142 question no. 2). And further point to the Siddur Qavanoth ha-Ari as endorsing the practice. Some also cite the precept that the holiness of the night of Shabboth renders it לילה כיום יאיר (a "night that shineth like the day") and thus the general concerns at play during the weekday do not come into effect. There are also those that in an attempt to reconcile the kabbalistic opposition to talleth at night with the practice of wearing it on Friday night, lower the talleth from their head to their shoulders (I'll cite the source for this later, time permitting).
Another aspect of kabbalistic consideration is, according to some sources, the spiritual prowess of the wearer. The Osar ha-Haim: Minhage Rabbenu ha-Qadosh m'Sanz (p. 198) reports in the name of R. Mordekhai Matili that it is only appropriate for those that are capable of effectuating a tiqun nefesh (cosmic rectification of souls) to wear a talleth for Qabbalath Shabboth:
רבינו שאל פעם את חתנו הרה"ק ר' מאטילי ז"ל מדוע אינו לובש טלית בליל שבת
קודש לקבלת שבת וקידוש כמנהג צדיקי זמנינו והשיב לו שראה בספה"ק שמי שאין
בכוחו ואינו יכול לתקן נפשות ישראל לא ילבוש טלית לקבלת שבת ע"כ גם הוא
אינו לובש אמר לו רבינו שמי שתיקן את נפשו רשאי וצריך ללבוש
In light of all of the above, it would seem that even among those sensitive to kabbalistic considerations, there is no definitive one answer as to the propriety of wearing a talleth on Friday night in order to greet the Shabboth. Rather there are many competing traditions on the matter.
