1) Sodium is an essential nutrient so you need to consume it regularly. The safe minimum sodium intake for individuals who do not sweat excessively is said to be 200 mg sodium (500 mg salt) per day.
Recommended Dietary Allowances: 10th Edition, 1989, minimum intake for salt:
In consideration of the wide variation of patterns of physical activity and climatic exposure, a safe minimum intake might be set at 500 mg/day. Such an intake is substantially exceeded by usual diets in the United States, even in the absence of added sodium chloride.
2) When sweating a lot, you may need more than 5 grams of sodium per day:
Sweat rate and sodium loss during work in the heat (PubMed, 2008):
People working in moderately hot conditions for 10 hours on average will lose between 4.8 and 6 g of sodium (Na) equivalent to 12–15 g of salt (NaCl) depending on acclimatisation.
3) What can help to prevent kidney stones?
According to Nutritional Management of Kidney Stones (Nephrolithiasis) ( Clinical Nutrition Research), these dietary measures can help reduce the risk of calcium kidney stones:
- low in sodium (2-3 g sodium/day)
- high in potassium (in fruits and vegetables)
- high in calcium (at least 1,000 mg/day)
- low in animal protein
- only moderate intake of vitamin C (<1,000 mg/day; avoiding vitamin C supplements and fruit juices as a source of fluid)
4) Treatment
Diet does not likely cure the established kidney stones. A doctor can tell what is the optimal treatment for a given type and size of stones.
To answer more directly:
Can the body rely on some sodium reserves during the time it takes to get rid of the intruder? (meaning no salt diet)
Most healthy people have so much sodium stored in the body that could probably be fine without any sodium intake for 2-3 weeks or a month as part of total water fast (I know few people who did it). But this cannot be made into an advice, because there can be some rapid sodium loss through sweating, so zero-sodium diet is not an option. Anyway, such a short zero-sodium diet would very unlikely have any meaningful effect on kidney stones. And, again, diet does not cure the stones that have already developed.
Is it better to take a minimum of salt daily?
In the context of preventing kidney stones it is enough to reduce sodium intake to 2-3 grams (which is 5-7.5 g salt) per day, as mentioned above under 3) and further explained in the linked article.
Reducing sodium intake as the only dietary measure may have no meaningful effect, so this may work better as a whole approach by also taking care about consuming enough potassium and calcium and avoiding excessive intake of animal protein and vitamin C.
So, such approach can be somewhat preventative, but the effect can differ a lot from person to person. This answer was limited to most common type of kidney stones - calcium oxalate stones. Other types of stones (struvite, urate and cystine stones need other approaches, so first it is good to get an exact diagnosis of the type of stones.