To whom do I need to pay my taxes?
To each country you'll be working in. In addition, you should check the laws of your country of citizenship, some have rules that define their citizens as tax residents unless they're tax residents elsewhere.
If you're moving to a different country every 6 weeks - you may end up with paying taxes to 8-9 different countries each year.
Keep in mind that many countries forbid employment for tourists, so you'll be very limited about where you can go with this. Unless you want to break the law, that is, but that is a dangerous game to play. Once you get arrested/deported due to an immigration law violation, you're likely to be deported to your country of citizenship and will require visas to many other places even if you're exempt due to citizenship (for example, if you get deported from any Schengen country, you'll be required to get a visa for any other Schengen country after that).
Also, in many countries you're required to register for, collect and pay VAT on services provided.
Do I need to pay my tax in the United States at all? If so, do I need
to pay Social Security and other social coverage? Do I need to pay my
tax to California where the company is located?
No, but you may end up being considered California resident even if you're not US resident for Federal tax purposes (as is the case when you're on F1 OPT).
What benefit do I get from having a "contractor" title as opposed to
"full-time" in terms of taxes? (All the perks such as vacation days
will be the same)
Depending on the country you're working in this may have different implications. However, it is very unlikely that you'll be able to pull this off as an employee since it will require your employer to register in all the countries you'll be working at and I doubt they'd want to do that.
Should I get paid to my personal savings account in the U.S. or in
another country? What are some benefits and restrictions?
Depending on the countries you'll be working at and the countries of your citizenship there may be some limitations. You'll have to check that.
To clarify certain issues with regards to taxes that came up in the comments:
Employment defined: providing personal services for a pay.
Income source: Income is sourced to the country where it is earned. Income is earned in the country where the services are provided. Services are provided where the provider is physically present.
Foreign income vs. Domestic income: Income that is not sourced to the country (i.e.: not earned for services you provided while being in that country), is considered foreign by that country.
General rules of income taxation: With the notable exception of the US, most countries tax everyone on income sourced to them, and residents on worldwide income. The US considers its citizens to be its residents regardless of where they physically reside.
Why is this different from a writer answering emails: writer doesn't provide services.
Why is this different from a 1%-er answering emails: 1%-er doesn't provide services.
Why is this different from a contractor answering emails: it is not.
Why contractors don't end up in jail for answering emails: tax treaties protect them in these cases. Most tax treaties between most countries exempt short periods of work from the sourcing definitions.
Why tax treaties won't help the OP: he plans on not being resident anywhere, tax treaties only protect residents of one country from the tax laws in the other.
Why people don't like hearing this (or similar) answers: because people don't like hearing that they have to pay taxes. Please don't downvote this answer if you're one of these people, its not my fault that you have to pay taxes.
With regards to immigration - talk to an immigration adviser/attorney for each country you're considering working in. For most - working without an explicit authorization would be against the local law. For definition of working see #1 above, however many countries (including the US) don't consider you being paid or not as a factor.