I am not a historian but I read a lot on the early history of Republic of Turkey so I wanted to write what I know since there may not be many people specializing on Turkey here.

When Sultanate was abolished (in 1922) and the last Sultan, Vahdettin went to Malta because he felt unsafe in Istanbul, the parliament had a closed session where they discussed how to proceed. I went back and read the transcripts. They discuss whether British will try to use Vahdettin as caliph (which seems to be a real concern), whether caliphate should stay in Istanbul (capital of Ottoman Empire) or brought to "Anatolia" (probably referring to Ankara, the capital of the modern republic but Bursa is also mentioned), who to choose, etc. Government suggests keeping it in Istanbul and electing Abdulmecid as the new caliph (There are a lot of discussions about whether a fetva is needed for these actions etc).

During these discussions, one of the MPs (Yusuf Ziya) says something to the effect of 'caliph shouldn't be just a title, he should have some authority, we are not imitating Vatican Palace, which is a surprising reference. Ataturk dismisses these concerns. There is a lot of discussions about whether the new caliph should swear in the parliament, how should he notified, how should the ceremony organized (religious or secular etc). There are a lot of religious concerns brought up here.

In the end, they chose Abdulmecid and send a delegate to notify him. He wanted a large ceremony with traditional cloths, some additional titles, a literal throne etc. He didn't get most of what he wanted but in the end, the ceremony in Istanbul had a lot of supporters of caliphate, which was disappointing for Ataturk and other secular leaders but it was also a sign that they couldn't just abolish caliphate.

Another thing to consider is that the support from Indian Khilafat Movement was disturbing as they were forcing UK to take actions against Turkey while the government was trying to negotiate a peace agreement. It looks like this played a major role in the end: When the government was working on the budget for 1924, Abdulmecid asked a higher allowance than the last year and it was already higher than the allowance of the Head of Republic or President (Ataturk). However, another request was that he be allowed to have meetings with diplomatic guests. After hearing news, Ataturk wrote to Ismet Inonu telling that Abdulmecid's extravagant lifestyle was already causing problems, that he was getting paid to be able to survive, his title is just a memory and the fact that he wanted diplomatic meetings was an attack to the republic.

Ataturk invited some journalists (apparently, they were sued as caliphate supporters in some sort of make-shift court a few months back) and said we are thinking about abolishing caliphate soon after these requests. He gave an interview with a French magazine and said "We have been happiest when our rulers weren't caliphs... Caliph has never had the power Pope in Rome had on Catholics... We kept it in place as we respect an old and laudable tradition. We respect him."

Days later, Abdulmecid met British representative Lindsay and discussed political matters. At the same time, Ataturk, Ismet Inonu, Kazim Karabekir (people in Ataturk's inner circle) had a meeting with some generals to discuss whether Mussolini could use Greece to attack Turkey and how might that work out. There, they also discussed caliphate issue and how to abolish it and it seems the meeting with Lindsay was used as a way to persuade the more religious among themselves. 

A few weeks later, when parliament had a session on the budget, some MPs proposed abolishing it as it caused conflict. Two MPs opposed it (but only one of them voted against the law). Surprisingly, many of the religious MPs actually supported and talked in favor of abolishment.

In general, in most of Ataturk's actions about Islam and Secularism, you can see a huge difference between what he did publicly and in private. He would recite prayers, verses from Quran etc. in the parliament and then at the dinner table, he would talk about how Turkey needed to be a secular republic. We know these thanks to memories of his closest allies. (Falih Rifki Atay's Cankaya is the best known account of those days but there are many other memoirs that are very helpful in understanding what he actually thought.) Overall, it is all part of a progress in their view from Sultanate to a modern, secular state. Looking at the whole picture (from 1920 to 1938) I wouldn't say that Ataturk ever considered keeping caliphate in the long term but there were many religious MPs and ministers who weren't on board with abolishment. In fact, many historians and social scientists suggest that this was one of the reasons for having an election in 1923: They more or less saved the country at that point so they didn't need support from everyone. They replaced some MPs they considered dangerous. The new members were much more open minded. (I think Competitive Elections in Developing Countries by Ozbudun was a good source on this.) Actually, if I am not mistaken, the only person who joined the parliament in 1923 that Ataturk didn't personally choose was also the only person to vote against abolishment.

Sources: (Unfortunately, all of them are in Turkish.)

Mete Tuncay, Türkiye Cumhuriyeti’nde Tek Parti Yönetimi’nin Kurulması, 1923-1931 https://www.tbmm.gov.tr/tutanaklar/TUTANAK/GZC/d01/CILT03/gcz01003140.pdf Ali Satan, Halifeliğin Kaldırılması Ömer Kürkçüoğlu, Türk-İngiliz İlişkileri http://www.radikal.com.tr/yazarlar/ayse-hur/mustafa-kemalin-altin-vurusu-halifeligin-ilgasi-1200340/ Falih Rifki Atay, Cankaya

Hello! My name is Payce, and I am a current student in college actually studying the first part of your question! Let me try to give a good explanation here, as well as some good material that may be of interest to you! Overall, it is possible the consideration of taking over the Catholic Church took place, but it was more fruitful for them to create a working relationship with the Vatican instead, and the granting of Vatican City as its own city-state was a part of that. Please forgive me, as this may need some framing.

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Mussolini and his Fascist movement needed to rise with popularity. See, a key feature to actually forming a Fascist government for Mussolini and Hitler was the fact that they needed to be popular and legitimate. Mussolini for example played the role of being seen as a man who would "fix any problem, no matter how small". He seemed like a man who'd send someone to fix a pothole for you if he got your letter, and Hitler caused many of the rural communities to get electricity for the first time. This video actually shows that sense of Mussolini being a man of the people pretty well, its a 1930s ad from Fascist Italy: https://www.youtube.com/watch?v=ZtTCHQCDlNU

Now, beyond that, a key feature to gaining popularity in politics is the appearance of being the moderate. I know, Fascists as the moderates? Its crazy! However crazy, it also was their goal. Two examples of this are the moves that are explained by Professor of Italian Studies, Christopher Duggan, in his book Fascist Voices. He notes that the Fascist party merged with the Nationalist party in February 1923^(1) , which made them seem like the party that would work more towards a middle ground than others, and also the fact they spoke in the language of the Risorgimento (the previously popular party that led to the Italian states unity and liberty, AKA unification of Italy and gave them freedom as a nation), and "...spoke of how their commitment to fascism emerged from a belief that they were finishing the uncompleted work of the Risorgimento"^(2), which made them seem like they were the Risorgimento reborn. This is all said to show how they worked in that pursuit of seeming to be the moderates and the party that people could turn to, since the opposition were the Socialists, and not quite everyone wanted a Socialist revolution like them, especially the Catholic Church, since the rebuking of the Church was a piece of Socialist ideology in Italy.

Now, let me address your question. Why let the Pope and the papacy have the Vatican? Well, it all goes towards that look of being the moderate. If Mussolini had torn down the Catholic Church's own workings, it would of killed not only his credibility with the nation (most Italians were Catholic), but also externally. Every Catholic would've looked at this and began working against Fascism, instead of seeing it as what he tried to make it, his ally.

Allow me to explain.

Mussolini knew that he wanted, and kind of needed, the support of the Catholic Church to add legitimacy to his regime, and he realized that interests may of actually aligned more than differed. Duggan notes that "...so many clergy had served, often with great distinction, as chaplains in the army during the First World War...strengthened patriotic feelings among Catholics and priests and hastened the erosion of the Church's traditional hostility towards the Italian state."^(3). The Church had actually become more aligned with them than the other rising party, the Socialists, which only became more so when the Fascist party and the Nationalists party merged. So beyond that, why else did they decide to support Italian Fascism? Well the Pope and Mussolini also saw each other as support for achieving mutual goals. The Pope influenced the masses, while Mussolini influenced politics. Both of them actually despised democracy and Communism/Socialism, and they held similar views and values, also, as part of their scheme to seem as moderates, the Fascists were willing to say you could worship both God and the Duce. Evidence of this is in this other propagandist film of theirs, more specifically at time marker 2:48, https://www.youtube.com/watch?v=qaDsHpwEAO8 . Notice they said they first honor their "mighty God", and then go into their Fascist notions more. This is an example of how the Fascists didn't mind giving the Church their part, and the rest of the day be Fascist material that was now backed by the Church. The Fascists seemed like they would play ball with the Church, and in exchange, the Pope actually played ball with them back. After all, it was in 1929 when the Vatican was given its statues as its own City-State and $92 million (more than $1 billion today) in compensation for the Papal States^(4). Basically it was a trade off in the existing working relationship for them to recognize the Vatican City as its own independent City-State and give them some money in exchange for the Papal States, as it worked more for them than simply overthrowing them and ticking off a lot of Catholics locally and globally.

Overall, the alignment of values over other parties, and the willingness to work together to dominate the masses and the political sphere led to a working relationship overall. There was give and take on both sides. None of this is to be negative to the Catholic Church, just is a scholarly reflection on history. I hope this helped with your first question, and I hope I can answer any further questions. Also, the book by Christopher Duggan is very well written and is a fantastic piece to study the history of Fascist Italy. Good luck in all endeavors!

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^(1)Duggan, C. (2012) Fascist voices: an intimate history of Mussolini's Italy. Bodley Head (Random House), London, pp352. ISBN 978184792103, pg. 67

^(2) Duggan, C. (2012) Fascist voices: an intimate history of Mussolini's Italy. Bodley Head (Random House), London, pp352. ISBN 978184792103, pg. 51

^(3) Duggan, C. (2012) Fascist voices: an intimate history of Mussolini's Italy. Bodley Head (Random House), London, pp352. ISBN 978184792103, pg. 82

^(4) Klein, C. 10 Things You May Not Know About the Vatican. History. Accessed on 22 April, 2020. https://www.history.com/news/10-things-you-may-not-know-about-the-vatican