Hello, and thank you for using JustAnswer again. I apologize for the delay in answering: this is quite an involved question, but as no one else has opted to assist you, I will try to address each of your questions briefly, based on my experience managing reception, call centers and customer service.

1. "How do I handle callers without being rude when I am alone and have several calls waiting?"
The key is to respond to each new call promptly, meaning to answer by the second ring, using a polite greeting to place each caller on hold. Even if a caller has to wait, as long as their call is picked up quickly, that initial acknowledgment will help dispel negative reactions.

Typically, this is something like:
"Hello, thank you for calling _____________. This is __________ (your name). May I please put you on hold?"

This is fine if you have one or two calls on hold which you anticipate resolving within 2-3 minutes. If longer, or more calls, also offer to take the other caller's number and call them back so they do not need to wait on hold, or if you have voicemail, offer to transfer them there to leave a message.

2. How can I give excellent customer service if I have to rush through my calls?

I suggest, rather than rushing through calls, if at all possible, to keep callers on hold appraised of wait time, explaining that you are assisting another caller but will assist them in turn, and help each thoroughly. If this is impossible, then offer to take messages and return calls. If you do not feel these are options, perhaps give me some more context so we can troubleshoot.

3. What, specifically, should I say to the caller I am speaking to when the emergency phone rings, and I have to answer the emergency phone on the first ring? (I have to answer two emergency phones.)

Say, "I apologize but the emergency line is ringing, may I put you on hold please?"

4. What can I say to a manager from another department who complains that the phone rang too many times before I answered?

Apologize to that manager, and let him know that you are working to reduce ringtime to two to three rings maximum. Don't feel defensive, be proactive, and don't take it personally.

5. Are there other solutions to this problem?
The techniques I've listed above should help solve the problem, though if you need to provide more context, I may be able to help you troubleshoot further.

Please provide a positive rating so I receive credit for assisting you, and bonuses are always greatly appreciated.