More of a comment, but I need the space.
Firstly, are you sure about the temperature of the ice being minus 5° C? That's just below freezing temperature and very close to melting temperature.
Using the assumption that the ice would be a crystalline solid analogous to a crystalline rock such as basalt, it would be best to keep the size of the openings within the ice small. Make them as large as they really need to be. This will reduce stability issues.
Making the tunnels and the caverns can be done via three methods:
- Conventional drill and blast - not recommended because of issues
sourcing explosives and the damage the blasts will do to the walls. When blasting there is always a blast damage region around such openings. In rock, this can be between 2 and 5 meters, depending on the type of rock and the explosives used.
- Some for of grinding technology, such as tunnel boring machines or continuous miners. These would be less damaging to the walls of any openings.
- Some form of thermal tunneling where heat is used to melt the ice. This may need to be devised. However this method may erode the floors of the tunnels with the melt water. Edit, this may be involve using hot water jets to cut the ice, or even hot grinding, where some of the ice is sacrificed to create hot water to heat a grinding head. This would soften the ice & enable the grinding head to use less power.
One problem I see in having tunnels and habitats in ice how to overcome erosion of the floors of the travelways. In conventional tunneling and mining this is usually solved by continually maintaining the roadways with new gravel and grading with either a grader or the bucket of a loader. Having a bed of ice will be problematic, particularly one that always traversed upon.
Edit 21 June 2021
To get an idea of what might be possible regarding sub surface habitats on Mars and elsewhere, with some adjustments made for atmospheric issues, the opal mining town of Coober Pedy in South Australia offers a good example.
Edit 23 June 2021
Following on from comments made so far, a fourth option has been discussed: using diamond wire to cut blocks of ice. Given ice would not be as hard as rock, diamond in the wire could be replaced with tungsten carbide.
Edit 11 July 2021
Recently I watched a video which briefly mentioned the former "under ice" base the US military had on Greenland.
The construction of the in-ice base used in Greenland during the late 1950s to mid 1960s was different to what is proposed by this question. Trenches where cut into the surface of the ice covered, prefabricated buildings were placed in the trenches which were then covered with arched roofs.
The base was designed to be used for a period of 10 years, with suitable maintenance of the trenches. The trenches were occupied for 5 years and abandoned after 8 years. Trench deformation was a major problem, as was the location of the sewage sump, but that's another matter.
Part of the problem for the ice base was the movement of the glacier was faster than anticipated. This may not be an issue in your situation but the properties of ice will be.
snow and ice are viscoelastic materials, which slowly deform over time, depending on temperature and density.
Maintaining a cavity within a block of ice and ensuring its integrity (size and shape) will be more difficult than maintaining a similar cavity in competent rock.