Attached is your config with recommendation. Pls read on the NAT, conduit and ACL.
For router config (but no config here), it depends, but I think the config should be ok.
For the proxy server and its new public IP, allow only Proxy to access internet and deny others. Mapped this server to the public IP, as follow:
static (inside,outside) xx.xx.xx.10 192.168.1.20 netmask 255.255.255.255
Maintain the NAT, but use ACL on inside interface to control internal hosts access to internet, specifically for DNS query only:
access-list inside permit udp any any 192.168.1.20 any eq 53 --> permit internal hosts only for DNS server query outside your network
access-list inside permit tcp host 192.168.1.20 any eq www
access-list inside permit tcp host 192.168.1.20 any eq https
access-list inside permit tcp host
access-group inside in interface inside --> apply ACL to inside interface
Hope this helps. Pls rate all useful post(s).
AK