This is going to vary very much between different countries based on their local regulations.
It's certainly possible to open a bank account remotely, but you can't just open any old account. In almost all cases the bank will require you to appear in person, and even then many countries restrict accounts to local residents of that country only.
To open an account remotely, you are best placed to look for an "offshore" account. These are frequently used by expatriates who require a bank account in their home country after moving abroad and can no longer hold a "normal" account (or don't want to for tax reasons). Most of the time they will offer a range of currencies (GBP/EUR/USD being common) and you can hold multiple currencies if needed.
Offshore accounts are often based in unique geographical destinations (e.g. in Europe there are many offshore banks based in the Isle of Man, Jersey, Guernsey). A lot of these banks do accept remote applications to open an account, but the process takes some time and requires a lot of paperwork to be sent back and forth.
Regarding the minimum balance though, you'll find this is common. Offshore banks cost more money to run and a minimum balance of US$5000 is very normal. If you search around enough there are some with lower balance requirements, but your choices will be limited.
Wire transfers are certainly possible, but normally have to be via SWIFT (read: high transaction fees).
If this is too expensive for your purposes, your only other option is to take a trip to the country in question and go to a branch in person to open a "normal" account. Make sure you check that the banking rules of that country allow non-residents to open accounts before you travel though!