SLiRP is better for your first use. Let TAP installed, you can use it
later as eth1 and fast host only access.

I also found SLiRP to work where TAP failed  (in getting internet working -- I still use TAP to access the host).  You should be able to use dhcp on SLiRP, that works perfectly for me.

Are you connected to any VPNs?

Try resolving a host in windows, then pinging it in colinux with the ip.  Does that work?