It can be difficult to get answers from Amazon, so it's definitely smart of you to post here. I used to sell on Amazon myself, and I now work for ByteStand. We're the experts on combining Amazon and Shopify (which can get complicated, especially for folks like you who are based outside the US). So here are the answers to your questions:
1. I'm not sure if by "box" you mean the actual product packaging or the shipping box. If it's the actual product packaging, Amazon will put that box into an Amazon shipping box and send it to your customers that way. If Amazon is fulfilling the order, the product will be put into an Amazon shipping box. So don't expect them to use your company branded shipping boxes.
2. Delivery times are kind of all over the place right now, but it's slowly getting better. It really depends on the product and where the customer is located. If the product is in stock in a UK warehouse and it's being shipped to a UK-based customer, it shouldn't take more than about 4-5 days for standard delivery.
3. FBA and MCF are two different Amazon programs. FBA is for orders placed on Amazon and MCF is for orders placed on a different platform (like your Shopify store). You can read all about how MCF works in this article, but the main thing to note is that MCF fees are generally about 30% lower than FBA fees (no referral fee). I'm not sure about their UK-based MCF fees, but here are their US-based MCF fees. Those fees are the total amount Amazon will charge you to ship your product to the customer. If you don't want to deal with figuring out how much it will cost to ship each product (and combination of products) on your store to all the different locations Amazon ships to, you can use the FBA Shipping app. It automatically calculates Amazon's fees at checkout and passes that cost on to customers as the shipping rate. You can also modify it to offer free shipping or a certain amount over/under Amazon's fees.
Hope that info helps! Let me know if you have more questions.