Can we convert two type of USB ports by simply re-mapping the pins?
USB type A, B, mini-B (also called mini-USB), and micro-B (also called micro-USB), are electrically compatible, and have the same four pins, each in a differently-shaped plug and socket. (The mini and micro connectors have an extra fifth pin, which can be grounded inside the cable to signal to the device that it should operate in "USB-OTG" mode, as a host instead of a peripheral. If you're not using that mode, you can ignore that pin; if the device or the cable does not support USB-OTG, the pin will not make contact with anything.)

USB 3.0 added new styles of connectors with 9 pins, of which the first four pins are electrically compatible with USB 2.0, and the other 5 are for new signals. The USB 3.0 A connectors and sockets are physically and electrically compatible with USB 2.0 A connectors and sockets; the extra pins will simply not make contact if they aren't used. The USB 3.0 B and micro-B sockets are physically larger than the USB 2.0 versions; they will accept either the old or the new plugs, with the extra pins being in a new part of the connector that is left empty when an old plug is inserted. (There is no compatibility the other way; a new plug cannot be used with an old socket.)

(I can't find information on whether there are USB 3.0 mini-B plugs or sockets, or whether USB 3.0 micro-B has the extra USB OTG id pin.)

USB type C, on the other hand, has 24 pins, and is a wildly different, much more complex beast. Even making a backwards-compatible C-to-A or C-to-B cable requires special handling: a specific resistor value across specific pins to identify the type of cable. Working directly with USB-C connectors in a project is not a good idea unless you've studied it enough to know what you're doing; my impression is that most applications use off-the-shelf USB-C controller chips to handle the complexity of the spec.

More information about USB 3.0 connectors: https://en.wikipedia.org/wiki/USB_3.0#Connectors
More information about USB type-C connectors: https://en.wikipedia.org/wiki/USB-C