Hi Veedushi,
As mentioned above, the crisis is not yet over in the Philippines, especially in Manila and Cebu City. As you might imagine in a developing country there are crowded areas that frequently do not have adequate water and sanitation for proper virus hygiene. There may also be many people living in one or two rooms, which makes the spread of the virus inevitable. Since there is not much of a public safety net the situation is very difficult for those without work at this time. Many are in dire straits.
I will see if I can answer your questions although please understand the situation is fluid and quarantine levels are changing every few weeks in some areas. Note that I am located in Cebu City so do not have as much information for Manila area - hopefully others will correct anything I may have missed or add specifics for their area.
What are the current regulations for entering Philippines? (quarantine, health checks, fees involved…)
No one can enter the country except returning nationals, spouses of nationals, or those with a Long Term Visa. Those who do are required to get tested and quarantine for two weeks upon arrival to the country. It is a fairly involved process to get back in as other posts describe in more detail.
Currently most of the country is still under GCQ - General Community Quarantine in an effort to contain the virus. Cebu City currently is under a more restrictive MECQ - Modified Enhanced Community Quarantine due to a surge in the virus in June when restrictions were first eased.
Info on Quarantine levels:
In Cebu City, one of the hardest hit areas and a popular tourist destination, a quarantine Pass is required to leave ones immediate neighborhood to shop, and 26 barangays are under a specific (granular) lockdown due to outbreaks. In those areas food is brought in and residents cannot leave.
Also those under 21 or over 60 are not allowed to go out except for essentials, and one cannot transit from Cebu City to the surrounding areas, in order to prevent spread of the virus. It is hoped (no guarantee) that Cebu City will ease to GCQ on Aug 1.
* Note - Nearby Mactan Island, where the airport is located, is under less restrictive GCQ.
Have there been visa changes recently? (new visas, cancelled visas etc…)
Visa upon arrival and general Tourist Visas have been cancelled since March 2020. Only Residents, Spouses of Residents, native born Foreign Nationals and Foreign Nationals with long-term Visas (issued under Section 13) are allowed entry at this time (July 2020).
Is it easy to find work in Philippines following the crisis? (working conditions for foreigners, quotas, new areas of work, incentives for remote working…)
It is never easy for foreigners to find work in the Philippines and it is worse now with so many locals unemployed. Many people have no source of income and without a government safety net the situation is dire for those individuals.

How would you view the local healthcare system in light of the crisis? (response to the COVID-19, how to get tested for COVID-19…)
In some of the major areas, especially Cebu City, the hospitals are currently at maximum capacity. Testing for Covid is required upon admittance to a hospital for any reason.

In Cebu City testing is available at Vicente-Sotto Hospital.
Has anything changed regarding universities and schools? (safety for foreign students, changes regarding student visas or conditions…)
Currently elementary schools will be re-opening with distance learning on August 24, with in-person classes tentatively planned for January 2021. Same with private schools. Some colleges are holding in-person classes in areas with low risk.
How is the real estate market following the crisis? (price, availability…)
Not much change that I am aware of.

Has the cost of living changed because of the crisis in Philippines?
Not so much that I am aware of but remittances from OFW are down, which has lowered income for a percentage of the population.

Is there information relevant to °µÍø½ûÇø retirees? (new measures for the elderly, visas for foreign retirees…)
In most areas under GCQ (General Community Quarantine) those over 60 or under 21 are not supposed to go out except for essentials such as groceries..

How about lifestyle? Have there been major changes in habits following the sanitary crisis? (measures in public spaces, new habits to adopt…)
Of course. Major malls are still operating at low capacity in most areas, especially Cebu, masks are required in most public places (5K fine if caught without) and banks and other buildings have instituted temperature checks. In Cebu City, a quarantine Pass is required to leave ones immediate neighborhood to shop, and 26 barangays are under (granular) lockdown due to outbreaks.
Of course always check reputable sources for the latest information.