Were they allowed to remain together?

As an ethnic entity - no, as families - sort of yes. Sonvarkom's executive order from 28 december 1943 states that kalmyks are to be evenly distributed between four destination points across Siberia - Altaiskii krai, Krasnoyarskii krai, Novosibirskaya oblast and Omskaya oblast. Several thousand were then relocated further north to Tobolskii, Khanty-Mansiiskii and Yamalo-Nenetskii okrugs.

Families were allowed to stay together as a household unit. NKVD instruction for the execution of operation 'Ulusy' (codename for deportation) states that non-Kalmyk women married to Kalmyk men are subject to the deportation while Kalmyk women married to non-Kalmyk men are not.

People were brought to the collection points on the territorial basis for further transportation. Personal accounts from Ubushaev's book tell that treatment of people varied: some were allowed to prepare and gather essential belongings, some were not or that some people's baggage was discarded to make space for more people or expropriated by NKVD.

The condition of transportation was extremely terrible, especially given that the deportation was happening in the middle of winter. Personal accounts tell about frostbitten dead bodies already at the collection points due to long waiting times. NKVD official reports, also presented in Ubushaev's book, state about 100 deaths on average per train which carried about 2000 people with average 100 shortly after arrival due to poor accomodation conditions. Ubushaev also notes the discrepancy between reports of train captains and further NKVD reports.

Conditions during the deportation were very challenging as well. "Special settlements" they were placed into were basically labour camps with harsh weather conditions, poor housing, little rationing and grueling work requirements. The degree of harshness is illustrated by the dynamics of population count: in 1939 census there were 134 thousands of Kalmyks; NKVD reported ~100 thousand people deported initially in december 1943-january 1944; Official USSR statistics on population show that there were 61 thousand Kalmyks in 1959.

In March 1956 the status of "special settlers" was lifted but Kalmyks were not allowed to return officially. In January 1957 Kalmyk autonomous okrug was formed within its former territory except the parts that now constitute Astrakhanskaya oblast. The possessions prior to deportation were not restored but accomodation were made for Kalmyks to transport any accumulated possesions during the deportation. During return Kalmyks were able to express the wish as to which part of Kalmykia they wanted to settle back and given respective destination for their travel vouchers. Kalmykia was not extensively settled by others during the deportation, so returning Kalmyks basically started from scratch via the formation of collective farms, mostly cattle herding, as before the deportation.

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Sources:

Павел Полян - "Не по своей воле"

Николай Бугай - "Берий - Сталину: по вашему указанию"

Константин Максимов - "Трагедия народа" (и другие)

Владимир Убушаев - "Калмыки: выселение и возвращение"

So much like other deported peoples (Chechens, Crimean Tatars, Meskhetian Turks, Germans, and so on) the Kalmyks were effectively all moved in one large action, done in four days in late December 1943, with about 120,000 people being deported. They were sent to either Central Asia or Siberia, and the Kalmyk ASSR was abolished, its territory split up amongst neighbouring regions.

Once they arrived in their destination, it proved difficult. Many did not speak Russian, which made communicating with the locals, and did not know how to survive the cold winter of Siberia or Kazakhstan. Rumours that they were cannibals also spread through the locals (possibly deliberately by the NKVD, the secret police who deported the Kalmyks), so they were even more reluctant to help the arrivals.

At first many Kalmyks were housed with locals, and eventually were moved into their own housing, though it was decrepit and not of any decent standard. Food was not much better, with potato peelings and animal corpses being frequent meals. A traditional tea that had previously been used in times of famine was also brewed and drank. It's worth noting that their traditional diet was heavy on meat and milk products, and not so much on potatoes.

In terms of what they did, Elza-Bair Guchinova gives exact numbers for this, which I will quote: "different spheres of the economy, such as agriculture (28,107 people), mining and gold mining (1,632), coal industry (784), paper production and wood processing (259), and other new occupations (8,608)." She also notes that "the Kalmyks’ attitude to labor, often forced, differed from that of Chechens and Ingush, who were more defiant" (in that the latter refused to do work as a protest). In contrast Kalmyks worked a means to show their loyalty.

After a few years children were allowed to enroll in school, though they were never praised or rewarded, even if they did well; they were also not allowed to receive any post-secondary education. This was in part due to their restricted living situation: they could not leave their area without a government permit, and as "enemies of the people" that was nearly impossible to receive.

By 1955 the rules were relaxed a bit. The deportees were allowed passports, and in 1956 they were granted some modem of movement and surveillance ended. A Kalmyk oblast (province) was established in 1957, and upgraded to its old status of autonomous republic the following year; the re-establishment of a Kalmyk state also meant some could return, which they did.

Upon returning to their homeland the Kalmyks felt stigmatized, with some returnees later claiming the word "Kalmyk" itself had become "obscene." However there is not much else written about them upon their return, so I'm going to have to leave it here.

Most of this came from Elza-Bair Guchinova, "Deportation of the Kalmyks (1943–1956): Stigmatized Ethnicity", a chapter in Empire, Islam, and Politics in Central Eurasia (2007), edited by Tomohiko Uyama. The whole book is available online if you're curious to read more, with each chapter a separate PDF.