|In my opinion, the procedures (in the context of this question, though the variations are there) are common for all irrespective of caste and creed.
Especially in Tamilnadu, it is just common for all Hindus, but slowly this has degraded in to brief two categories. What to do on which days is detailed in depth in Garuda Puranam which is a FAQ on these aspects asked by Garuda and answered by Lord Sriman Narayanan.
These practices are practically not possible to follow even for the most rigid followers. For example, there is a ceremony called sothakumba which has to be performed on daily basis for the first year. This in present day is not practical and hence to avoid total non-observance it is been practiced once a month.
Generally speaking the relaxed observance are seen among brahmin community. For this community and with the relaxed observance procedure, the days 1-13, ~30, ~45 ~180, ~360 are significant for the first year and one day in subsequent years.
This has fallen down to 1, 10, ~30, ~360 among non brahmin community. Though there are no such relaxations prescribed, many in Tamilnadu follow this pattern.
As far as the significance for these numbers, they are: 1-10 are the days where the departed soul will get a sukshma sareera (body) and the body grows and get organs based on the pinda (rice balls) that the kartha offers on each day.
11-13 are the days which is called ekodhishtam, sapindakarma and ‘subha sweekaram’.
In the 11th day, the athma which got the new sareera is about embark a journey this day to other loka.
Vasu, Rudra, Aadhitya are the three layers of pitrus. When the departed soul starts its journery to Vasu group and it will have a cascading effect on other groups and finally the Aadhitya group will go out these three and may be born as another jeevan or goto swarga or naraka. This is done on the 12th day slicing the rice into halves.
The 13th day is the day where the grieving family is ready to accept the shubham and hence the name.
In this context the time for us is different for the time for pithrus and so in every other lokha. A human month is equal to a pitru day. Hence the kartha has to do the ceremony on every month. This is so mandatory for the departed soul as this soul will carve for this food. Being so important, just to avoid non-compliance by unforeseen events (like death for the kartha), it is been offered 16 set of stuffs like rice, vegetables, jaggery etc as donations.
The numbers mentioned in the example is relevant in case of the death. There are different types of pranas and jivas in our body. After the death of a person each of them leaves the physical body (or annamaya Kosa). When the death take place first of all pranavayu is leaving the body and depending on the behavior all of them specific time to leave the body. Also the soul too have it’s timings to leave the body. On 3rd day of death there are certain rituals to perform and these are done for the moving of these things smooth. The same is applicable to the 41st day.
The number 16 represent the 16th Sansakara ‘Anthyeshti’. There are 16 samskaras in total. There are 15 samskaras before the 16th one, like gharbhadana,pumsavanam,etc. So after the death of a person the rituals mentioned in the ‘Antheyshti’ will be done by relatives.
Note: Don’t have enough details on these. These are just my basic understandings and this answer is certainly not complete.