Good thing you separated those questions, the second one can be pretty daunting.
Since you put things under the heading FSMA, I will do my best to answer from that perspective.
1. The team question. In Preventative Controls, there is no mention of a food safety team the way BRC or other GFSI benchmarked programs describe it. FSMA talks about the "PCQI" or preventative controls qualified individual. You took the training, good start. There is nothing wrong with maintaining a "team" approach to developing your food safety plan (FSMA doesn't say it has to be only one person), and will that way satisfy both FSMA and other audit programs. What is important, is to focus on competency, not the numbers. Whoever is responsible for he food safety plan needs to have the ability and knowledge to design and maintain it across the facility.
2. Sending out your food safety plan. The part in preventative controls covering supply chain approvals and monitoring give options that range from audits to system review. Look up 21 CFR 117.410 for all the fun details. Essentially the requirement is that a manufacturer (who falls under the FSMA rules) have a risk based system to approve suppliers. You need one for your suppliers, your customers need one for you. Obviously I have a bias for using GFSI benchmarked certifications as the basis, but there are some instances where simply reviewing a suppliers food safety plan, or parts of it, may be sufficient. The challenge would be in the assessment - if there is insufficient evidence in the audit report that the auditor reviewed the food safety plan, the audit doesn't give you enough.
Here is the way to assess what the best course is. The Agent in Charge (that's site senior management) and the PCQI (thinking that's you) are responsible and liable for the food safety plan. You both need to be confident that the risk based supplier approval mechanism is going to stand up to critique if things should go wrong in future, since it's you two who are held accountable. I would recommend erring on the side of robust, not simple.
There is also the time factor. You would need to have the time to chase, receive, and assess (don't forget to document your assessment) of each plan. From an effectiveness, and efficiency stand point, having every customer ask every supplier to provide their food safety plan documentation is not good. There is also the document security aspect to consider when shipping things off site - and that goes both ways - if you are holding a load of customer information you need to make sure it's secure. The one core weakness in trying to approve a supplier based on a paper program sent in - you have no idea if it's a valid representation of the supplier.
If a customer demands you send it, and are unwilling to accept a recognized third party audit, you may be stuck with it. For your suppliers, use a more effective method wherever possible.
3. The broker question can be a really long and complicated answer.
First look at the role of the importer of record. This would be the US based person or company that is legally importing food products - they have the responsibility for supplier approval according to FSVP. It may be the broker, it may not - good question to ask. Supplier approval is similar for foreign and domestic, you need to verify the food safety plans of both.
If you have a good broker, they can have their own supplier approval program that satisfies FSVP (BRC has a great guidance document for brokers and importers of record). You have to essentially verify their supplier approval program.
If your broker doesn't have supplier approval programs or information, you have to see through the broker to the manufacturer. Either way, you need to know someone has approved the supplier, and using what methods.
Good brokers should be able to provide all the information you need to ensure foreign suppliers meet your risk based approval program.
Ryan had the right advice above - talk to the broker and find out how their supplier approval works. The last thing you want is just before a production rush, finding out your broker can't source materials for you because they, or the supplier, weren't ready.