Can I use both my internal and external graphics card in same system?

In PCI and PCIe systems, it is certainly possible. The BIOS has settings to say which is the primary, if both are present. It is subject to the usual issues of mixing graphics cards of different brands: the drivers may be allergic to each other. At least, be sure not to install the "utilities" from both as they may fight over things.

Before switching to PCIe (2008), the built-in graphics was mutually exclusive with the AGP slot, but it was still possible to add PCI graphics cards, and I did so when serious non-AGP cards were becoming hard to find.

The lingering memory of not having both is from AGP.

The flip side is that if you're not using the built-in graphics because nothing is plugged in to that connector, you should turn it off in the BIOS to free up resources. People are running both without knowing it.

A few notes as I ran into same setup recently (2 displays connected to 2 different GPUs, one integrated).

On my MSI motherboard, there is indeed a setting in BIOS (well, UEFI stuff, but you know what I mean) in section about graphic adapters that allows you to select whether the integrated GPU should be used at all (it says something about multi display setup). This way you can use on (main) display via "external" card, the other via the integrated.

Windows 10 don't seem to have any problem with this setup. Ubuntu 16.04, however, does. It fails to send visual signal to the secondary monitor connected to the integrated GPU (although the display is in use (as in the space for windows is there)).

Interesting thing is that when I forced one game (under Windows with the new AMD "external" card), which ran really poorly on the integrated GPU (back then when I had no external GPU), go fullscreen on the display I had connected to integrated GPU, it ran smoothly, obviously being rendered by the AMD "external" card. I think there could be some way through this to fix the Ubuntu problem mentioned above, but I haven't figured it out yet. I'll probably go with using a DVI->D-sub adapter and connect the old display to the AMD external card too.