Sharing a switch with both internal and external interfaces

Solution 1:

Congratulations. You've effectivley eliminated any security your router was providing for your internal network.

What you need to do is to put things back as they were and set up NAT on your router to NAT the appropriate public ip address to the appropriate private ip address.

Solution 2:

What would be any downsides to this setup?

Security. The machines to which you have assigned public IPs are now completely exposed to the Interwebs.

Would this compromize security on the network?

Yes. If your public machines get hacked you'll soon find you have problems on your not-so-private LAN.

Could machines access computers that only have a private IP assigned to them?

I assume you mean from the public network? No, they wouldn't be routable directly. But see above.

Anything else I should know?

Yes, go back to the original method. If it's a Cisco or HP variant you would setup NAT rules to map public IPs through to the internal servers, and then add specific access list controls to lock down the ports that are open, and control source networks if necessary.