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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 ...
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Several sets of rules (specifically the Schengen Borders code and the Free movement directive) interact here so it can indeed be a little confusing. A lot of material available online, when not outright wrong, is also concerned with the rules that apply to regular holders of Schengen visas or other documentation (i.e. people who are not married to an EU ...
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In principle, when presented with an Australian passport, border guards are supposed to look for an entry stamp to check you haven't stayed too long in the Schengen area.
In practice, they do not always check that very carefully and even if they do, I don't see how you would risk anything more than a lecture since, as an EU citizen your presence was legal ...
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You have to remember there are rules. Whether you like it or not, and whether this is fair or not is irrelevant, visas are not a right, they are a privilege. Each country (or here, union of countries, the Schengen Area) is free to decide who can enter, and under what conditions.
The rules are clear. You cannot ask for a visa to travel to country X when you ...
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Now do I need to apply for a new visa from Greece embassy for the same dates?
Not really, your visa is valid and it is not possible to issue two overlapping Schengen visa. What you could do is email the Greek embassy/consulate and asking them for advice. They probably don't care and will tell you that it's OK or dodge the question but it gives you some ...
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In general, there is some flexibility as long as there is no reason to suspect you lied specifically to circumvent the regular procedures and never intended to travel to Italy. It's even less of an issue for a multiple-entry visa (which is by nature not tied to a specific trip). See also Can I use my Schengen visa for a completely different purpose and entry ...
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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-...
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There is no such thing as a business or tourist Schengen visa. In principle, there are only two cases:
If your visa is a single or double-entry visa, it's tied to a specific trip and should in principle have a short validity. Once you have completed your trip you can apply for another visa.
If your visa is a multiple-entry visa, it does already implicitly ...
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