The best way to deal with the gatekeeper is to not deal with the gatekeeper if possible. Your best opener is "Hi. is John in?" or "I don't suppose John is in the office now is he? and nothing else.
Speak in your natural voice. Don't have a sales or telemarketing voice. This is a defense mechanism salespeople throw up to lessen the pain of rejection or failure and sounds unnatural and "salesy" to the person you speak to. Although sales isn't personal, talking like this makes it feel like they're rejecting your "phone persona" and not you personally. This is bad because it's unnatural and insincere and people will have less respect for you.
If you're getting rude responses it's because you don't sound certain of yourself. The person thinks you're wasting their time and will quickly take control of the conversation by backing you into a corner with questions like "what are you selling?" or "Is this a sales call?" Punch through the wall, not at the wall, if you're going to be doing it at all.
To lessen anxiety. BREATHE from your diaphragm (stomach). Take deep controlled breaths between every sentence as you read your script. This will work wonders.
If you get passive aggressive or patronizing questions such as "what's the catch?" "what's your pitch?" the best response is "I don't understand. What do you mean what's the catch?" Don't let them put pressure on you or defend what you're selling. Let them sound like an asshole if they ask again but most times they will back down.
If you get told to send them an email and you feel like it's them brushing you off, you can say "May I ask you some questions first to see if I can help in X area" then keep qualifying them.
If there's one objection which comes up over and over and over again such as dealing with an existing supplier you must PREEMPT this by bringing up the issue before they do. If you don't they won't see a valid reason to keep talking to you and you also lose control over the conversation when they interrupt with "but we're already using somebody else"
When I make a cold call, I say who I am and what company I'm from. I then say "I'm sure a company of your size/industry is already using X product or service, would that be a fair statement?" Then you tell them a story which implicitly shows them how not using your product/service is losing them money or opportunities i.e. a case study. If you don't preempt the objection the only thing the guy will be thinking about is "I have this sorted already, I'm busy, you're interrupting me, and I don't care. Bye."
A couple of phrasing tips here:
Instead of asking "How are you?" Ask "How have you been?" They will think you know them for a few moments which means you couldn't possibly be a cold caller They will listen a little closer to what you have to say next. Ask it sincerely like you expect an genuine answer from them.
Here's a tip I got from Claude: always ask permission to ask additional questions. Here's my twist on this. Make it to their benefit to answer said questions, too! "Would you mind if I ask you a couple of quick questions to see if I can help you get a better deal?"
A lot can be done to further a sale by eliminating the perceived risk and downside to buying from you. A significant part of my calls are reassuring the customer that buying is risk free. You can do this by mentioning there's no hidden fees, easy and simple application process, no sensitive information is asked over the phone, no lock in contracts etc. Basically whatever is applicable that will lessen the risk of them buying.
And a final tip: don't make it easy for them to hang up on you and get back to what they're doing. Do not ask "How does that sound?" (You'll always hear "send me an email" or "let me think about it" because that's a conditioned trial close they've heard countless times).
If you give them a choice between buying and getting back to what they were doing they will almost always choose the safer option even if they want what you have to sell.