Those are actually rather interesting questions. The completely honest answer is I don't really know. The man in the street will typically say "of course you need a work permit" but I am not so sure.
The practical answer is no, you don't need any kind of special visa or permit.
Generally the notion of a remote worker (a foreigner who is employed in their home country while physically being in other countries) fits very poorly with immigration/visa rules. Those things move at a glacial pace and have in no way kept up with technology (generally speaking).
The assumption underpinning pretty much everyone's work permit/visa rules is that a foreigner wanting to physically work in the local environment will be employed by locals ie they will participate in the local labor market, will take money from a local employer and be competing with locals for jobs. Alternatively you might be a visiting business person wanting to do business with a local company.
However I am Australian, I am legally an Australian resident, my employer is Australian, 100% of the financials (bank accounts, all paperwork) is in Australia and I pay tax in Australia. I have never taken a cent from a local business or individual, I am not participating in the local labor market in any way and I have no interest in doing business with local companies. I don't even have a local bank account or a local permanent address. Even if I registered to pay income tax in another country the amount would be $0 because I am not earning any money here. My work doesn't touch the local environment anymore than someone on holiday checking their email or taking a phone call from home.
So in practice, as far as the host country is concerned, I am simply a tourist. If anyone asks about my plans (which they very very rarely ever do) I am simply taking an extended trip (which is the truth) and I can show sufficient means of support (ie money in the bank).
As for the legality of this I am uncertain. I certainly don't have any ethical concerns. Were I to not do any work while visiting a country the local labor market would not be affected in any way, some locals would be deprived of additional income on which they presumably pay tax and a company back home would be deprived of an experienced worker if I was forced to not do my work while visiting other countries. No one is harmed by what I do and a lot of people derive some benefit from it.