You should renew the 1an1 certificate and buy a new one for the godaddy certificate from 1and1 - i don't believe you can renew certs from other vendors - although it wouldn't hurt to ask.
Once you get the new certs you can replace the old ones, your services should not be affected as long as the info is correct. There is no "Good On" type check for certs just "Good Until" checks.
SSL certificates are bound to the internal IP address of the web server, not the external IP addresses.
Let's say you have foo.example.com bound to Public IP A and bar.example.com on Public IP B, but your web server only has the IP address 192.168.0.1
Whether the request comes in on IP A or IP B, it is still going to end up at 192.168.0.1. Which means that IIS has no choice but to use the certificate that is assigned to foo.example.com.
To work around this issue, you will need to have multiple IP addresses assigned to your web server. This is easy to do. Speak to your sysadmin to have some IP's removed from the DHCP range (or ask him/her which ones you can use), then go to your properties for the network card (Control Panel > Network Connections), and go to the properties for TCP/IP.
You will need to have a static IP enabled in the first place (being a server I hope this is done anyway), and then click Advanced, and under the box for "IP addresses" click "Add" - and enter the new IP addresses you've been assigned by your sysadmin (Let's say 192.168.0.2).
Then, at your router, you need to ensure that requests from IP A on port 443 go to 192.168.0.1 and that all other requests on port 443 go to 192.168.0.2.
Then, in your IIS configuration, you need to bind the SSL Cert from foo.example.com to 192.168.0.1, and bind the rest to 192.168.0.2 (or leave as All Unassigned, as you have).
If this doesn't work, or you already have this configured, update your question and leave a comment to let us know.
Update: I just saw your comments, thanks for the update. You will need to ensure foo.example.com and bar.example.com are on two different public IP addresses. The reason being that because the packets are encrypted, there's no way you can use hostname based routing to send the request to the right IP address (I believe this is the case. If anyone knows different, let me know). The only part of the request that's visible to the routers is the destination IP. This is why you can only have one SSL per IP address. So you will need to have public IP's for this to work, and in your DNS an A record for bar.example.com that is different to foo.example.com.
Best Answer
The problem is that SSL negotiation is done before the host headers are sent, so you can only have one SSL cert per IP address. This is why you are getting the invalid certificate on b.mysite.com
there are two ways to handle this: