The topic of the question is not entirely covered by the Gemara quoted in OP. An important machloket between Rashi and Tosfot, well illustrated by Rabenu Yona on daf 5b must be included. I will try to answer the OP through Gemara and Poskim. If I am wrong, every criticism should be accepted and I will consider to modify the answer.
Two different topics.
The first, Gaava, as mentioned in the OP, is prohibited as an offense. This topic is mentioned in the title of the OP. It concerns activities which are not Avodat Hashem.
the second, non prayer activities , including Torah study before prayer, as a lack of readiness/risk to miss the starting of prayer.
We'll star by the second topic.
Gemara Berachot 5b:
תניא, אבא בנימין אומר: על שני דברים הייתי מצטער כל ימי - על תפלתי שתהא לפני מטתי... לא תימא לפני מטתי, אלא אימא: סמוך למטתי.
It has been taught: Abba Benjamin says, All my life I took great pains about two things: that my prayer should be before my bed ... etc... Do not read 'before my bed', but 'near my bed.
Rashi:
סמוך למטתי. כל ימי נזהרתי שלא לעשות מלאכה ושלא לעסוק בתורה כשעמדתי ממטתי עד שאקרא ק''ש ואתפלל:
... I have been careful to not work and not study Tora when I got up until I say KS and pray.
Talmide Rabenu Yona on Rif:
ומיירי בתפילת שחרית שאין לו לעשות שום מלאכה קודם שיתפלל... וכו' ... ורש"י ז"ל פירש דהיה נזהר דאפילו לקרות בתורה לא היה רוצה אחר שעלה עמוד השחר עד שיתפלל.
The Gemara talk about the morning prayer, that one should not make any work before this prayer. Rashi explained that he (Abba Binyamin) did pay attention not to read Tora after the early morning dawn.
ואומרים רבני צרפת ז"ל שלא מצינו בשום מקום שהלימוד יהא אסור.
But French Rabbis say that we don't found in any teaching that learning is prohibited.
ואפילו לדברי רש"י ז"ל אין לאוסרו אלא כשלומד לעצמו מפני שיכול להתפלל תחילה ולחזור ללמודו אחר כך.
Even according to Rashi, the prohibition concerns only a man who learn alone, because there is no problem to pray first and go back to his study afterwards.
אבל במי שמלמד לאחרים ודאי מותר כיון שהשעה עוברת דזכות הרבים דבר גדול הוא ואם לא ילמדו עכשיו יתבטלו ולא יוכלו ללמוד אחר כך.
But for one who teaches others, it is obviously permitted because this time is not recoverable. To give credit to people is something big. If they would not learn now their learning would be annulled, they would not be able to learn afterwards.
שכן מצינו לרבינו הקדוש ז"ל שהיה מלמד לתלמידיו וכשהיה מגיע זמן ק"ש היה מעביר ידיו (דף י"ג עמוד ב) על גבי עיניו וקורא ק"ש וחוזר ללימודו. ואחר כך היה מתפלל ולא היה חושש אף על פי שהיה מגיע התחלת זמן התפילה.
We found a such case with Rabenu Hakadosh of blessed memory, who was teaching his students, and when the time of kriat shema started, he was read kriat shema and continue to learn. Afterwards he was praying without concern for the staring time of Tefila.
ולא חיישינן שמא ישכח ותעבור השעה כיון שהתפילה דרך הוא להתפלל אותה בכל יום יתן אל לבו הדבר ויזכור ויתפלל.
There is no concern that he does not remember and until the end of tefila time because tefla is a quotidian habit. He surely would remember it.
Shulchan Aruch OC 89, 6
אֲפִלּוּ לִלְמֹד אָסוּר מִשֶּׁיַּגִּיעַ זְמַן תְּפִלָּה, וְהַיְנוּ מִי שֶׁרָגִיל לְהִתְפַּלֵּל בְּבֵית מִדְרָשׁוֹ, וְאֵינוֹ רָגִיל לֵילֵךְ לְבֵית הַכְּנֶסֶת, דְּאִכָּא לְמֵיחַשׁ דִּלְמָא מַטְרִיד בְּגִרְסֵיהּ וְיַעֲבֹר זְמַן קְרִיאַת שְׁמַע וּתְפִלָּה. אֲבָל מִי שֶׁרָגִיל לֵילֵךְ לְבֵית הַכְּנֶסֶת, מֻתָּר. וְאִם הוּא מְלַמֵּד לַאֲחֵרִים, אֲפִלּוּ אִם אֵינוֹ רָגִיל לֵילֵךְ לְבֵית הַכְּנֶסֶת, מֻתָּר, כֵּיוָן שֶׁהַשָּׁעָה עוֹבֶרֶת, דִּזְכוּת הָרַבִּים דָּבָר גָּדוֹל הוּא. וְאִם לֹא יִלְמְדוּ עַכְשָׁו יִתְבַּטְּלוּ, וְלֹא יוּכְלוּ לִלְמֹד וע''ל סי' ק''ו.
Even to study is prohibited at the starting of prayer's time. This prohibition concerns only those who are accustomed to pray in her Bet Midrash and do to go to synagogue. Those who go to synagogue can learn.
One who teaches others can teach even in Bet Midrash (outside of the synagogue)... (The following is quoted from Rabenu Yona.)
Mishna Berura opinion (OC 89, sk 32) , although it is somewhat different may be that the answer to the OP is: He can sit down and learn in a room where people are going to pray after a while, and people will remember him the prayer.
The first topic above cited.
Others activities (see bellow for nutritional activities) are prohibited, as we see in Tosfot witch discussed against Rashi opinion regarding learning only.
If the first Mynian is at a late hour, and he use to pray in this Minyan, see above the translation of the S.A., he should be sure to not forget and it is better to learn in a place where friends will see him.
If he was forced to make something before the time of the mynian, according to Chaye Adam it is better to pray alone before this activity.
Eating is obviously prohibited from Alot early morning dawn as mentioned on the Gemara, even if he has begun before. The source is the Gemara quoted in the OP, see SA.
But to drink water is allowed.
If someone does not feel well (very hungry, Coffee withdrawal syndrome), eating or drinking beverages other than water, most rabonim to drink Coffee without a great amount of sugar but normal sweetening (Mishna Brura) or sugar is like water.
If he feels weakness, he can drink a cup of tea or coffee with a slice of cake.
If eating will help to pray with more cavana, it is allowed anyways. Beur Halacha says that someone is forced to eat, it is better to say Kriat Shema before the meal. I skipped more details not linked directly to the OP.
What is called "Tefila" regarding the question?
For the problem of "אחרי גאיך " , main poskim mention only tfilat haamida,
For general occupations What may one pray to eliminate the problem? Rema mentioned that some poskim (Trumat Hadeshen 18) {See the Shut, it seems that the Trumat Hadashen prohibits Meyikar Hadin, but he has seen Medakdekin witch went about their daily lives after "Berachot Vetishbachot" }) allowed after birkat Hashachar (but he discourage this)
to eliminate the problem of "'ואותי השלכת אחרי 'גאיך"?
Note that Trumat Hadeshen do not include the topic of Gaava (eating and similar activities)}. Thus one may not " pray berachot" but tefilat haamida for "achsre geecha".
Shulchan Aruch did not refer to the kula of the Terumat Hadashen.
If an individual who rises BEFORE ALOS? a person may clearly recite certain tefillos starting at alos, but if it is still completely "night" by all measures? Let's say a person goes tobed after ma'ariv but rises at 2AM - what can/should he do considering that it is impossible to engage in the normal morning routines of tefilla?
Poskim clearly stated that he may make anything up to alot, even eating.
When the hour of Alot came, the problem begin to apply. So, it is possible to make anything before alot, following some interpretations on the Terumat Hadeshen words, that Birkot hashachar is sufficient to solve the problem. But the original purpose of the Trumat hadeshen is not Achare Gevecha, but the problem of Gemara daf 5b. I hope that this answer helps.
