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Can I travel freely (no COVID-19 restrictions) within Schengen Area after formally entering a Schengen country?
No. See https://reopen.europa.eu/en for the requirements of each country.
E.g. currently for France https://reopen.europa.eu/en/map/FRA/7001:
Non-vaccinated travellers arriving from Spain, Portugal, Cyprus, Malta, the Netherlands and Greece must present a negative PCR or antigenic test taken less than 24 hours before departure.
If you're vaccinated you should be fine in most (all?) EU countries, but this sometimes depends on which vaccine you had, and when you had it. E.g. Sinovac doesn't work for Italy https://reopen.europa.eu/en/map/ITA/7001
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There are even restrictions or requirements within a single country.
For instance, in France, any “long distance” bus or train travel (i.e. not commuter bus or train) requires you to have a valid “pass sanitaire”, which is proof of full vaccination, of a recent negative test, or of recovery (within a given time range).
You must have the recognised QR Code-like barcode to allow scanning by operators. So the first step for you is to convert your foreign vaccination certificate into EU format. A number of countries, including France, have a procedure for that. Note however that in general, only vaccines used in the EU are recognised.
In France you’ll also need the pass sanitaire for restaurants, museums, exhibitions, and larger malls and stores.