What can also help is specifying the right options right in ~/.ssh/config for certain hosts and domains: E.g. do "ForwardAgent no" globally, use a "Match *.my-trustworthy-company-domain.com" block and add "ForwardAgent yes" there.
Also very good for other options that are useful but problematic when used with untrustworthy target hosts, like ForwardX11, GSSAPIAuthentication, weaker *Algorithms (e.g. for those old Cisco boxes with no updates and similar crap).
Another neat trick is just using a ""Match *.my-trustworthy-company-domain.com" block" with an "IdentityFile ~/.ssh/secret-company-internal-key" directive. That key will then be used for those company-internal things, but not for any others, if you don't add it to the agent.
Whenever possible, yes, but AIUI it's not always possible; the one use case for which I believe full-on forwarding is required is using your personal credentials to transfer data between two remote servers (ex. rsync directly between servers). If there's a way to do that I would actually much appreciate somebody telling me, but I have looked and not found a way.
AFAIK, this doesn't solve the SSH agent problem - the problem is the agent has access to all of those keys regardless of the host you connect to.
So forwarding your SSH agent means an administrator of the system you're connected to could use any of those host keys loaded in the agent to connect to their associated machine.
Depending on what it's serving, and how up to date it is, and who else is on that network and can access the server, and who else can come into that same room when you're not there, and from where you get the software that you install on that server... it might be less trustworthy than you think.
Just use ssh-add -c to have the ssh-agent confirm every use of a key.