(1) Can I install the second VPN client alongside my current one? Or do I have to remove my current VPN software first?
depends a bit on the OS ; on Windows the tap-win32 adapter is installed. You can have only a single version of the tap-win32 active at a time. You can have multiple versions of the openvpn executables (openvpn.exe, openssl.exe, etc) at the same time - I actually run OpenVPN tests this way.
On other OSes the tap-win32 adapter issue does not come into play.
(2) Supposing that I can install two VPN clients from two different VPN providers, can I launch the 2 VPN clients to surf the internet simultaneously?
how would you do that? only one VPN connection can redirect
all traffic at a time, i.e. there can be only one default route. That's a networking/operating system limitation.
With some tinkering you could get both VPN services up and running at the same time, so that one is a fallback for the other, but this is non-trivial when using 'redirect-gateway'.