1) Are they an employee or a contractor? You mention "employees" but talk about them invoicing you. If they are requesting payment via invoice then it sounds like a contractor. It's very important that you define this relationship early on. There are different rules for both (health insurance, paid time off, taxes owed and by who) that are handled very differently. You might have seen a lot in the press about Uber / Lyft etc handling their drivers as contractors and a lot of these reasons are the reason why. So if you're deeming them as contractors (1099s) then make sure you have a contract that makes this explicit. You can find good templates from places like Upcounsel[1], but make sure you have a lawyer take a look.
2) I spoke with the founders of Deel[2] (YC W19) just last week. Their value proposition is that they'll handle the contracts, paperwork and payments (according to your payment schedule) for a small fee (I think $10/m/contractor). This seems like a very good value prop if you hire more than 1 or 2, and one I'm looking into.
3) If you have only a few contractors, then creating a standard contract and making payments to them on submission of their invoice and time sheet (if hourly) is pretty simple. I request our contractors only send invoices monthly to limit the admin for me, and then I pay them via Transferwise (they have a USD-USD beta program that I'm not sure is open to all), because it has cheaper fees than our bank.
4) Assuming you're using a fully fledged accounting/ payroll service or otherwise, then lots of these do have a contractor payment part. The payment part isn't the difficult part, the real benefit of these systems is handling the paperwork with the government for you. We use a full PEO called JustWorks[3] which is awesome - you tell them you have a contractor, you're paying them outside JustWorks and they'll handle all the year end 1099 forms etc.
5) Freelancers. We hire many freelancers to do work for us via Upwork. It's a great tool to get things done, and will do all payments/contracts/compliance for you, for cheap. The downside is the freelancer pays pretty hefty fees, which go down over time [4]. We pay $25/m for all our freelancers paid through the system (30+ freelancers), so it works really well for us.
Happy to answer any other questions you may have!
[1] https://www.upcounsel.com/free-legal-documents [2] https://www.letsdeel.com/ [3] https://justworks.com/features [4] https://www.upwork.com/i/how-it-works/freelancer/