I don't want to say, "It's worth stressing about," because anxiety is good for nothing, but it is worth preparing when you have a chunk of time to do so. Because of my busy work/work travel schedule and completing the last semester of my second degree, I largely broke the process into three chunks: visa, pet nonsense, packing/moving. Although there was some overlap (e.g., I've been throwing out/giving away/selling/donating my possessions for months), I think each one takes time and energy to get right. So it's good to get yourself in order whilst in a holding pattern on another part of the overall process.
Because I was flying BA, I worked with IAG Cargo on the US end and JCS on the UK end. IAG Cargo double-checked that I'd brought the customs form, a complete Annex II, acceptable certificates of health and their rabies certificates. I brought along a plastic sleeve for taping to each crate to hold the required documents for each of the cats, along with a photo of the cat who's supposed to be in the crate, and the rabies and microchip tags. The IAG people also remeasured the crates to confirm the dimensions I'd provided, did a security check on the crates, weighed them with the cats in, and attached all the documents for me. (Although they were super-disorganised and couldn't find their packing tape, and I'd luckily brought some with me, so I was able to pop to the car and retrieve it for use.) Then they took my money (cash only in Philly, terrible system) and all was settled. (Then I said goodbye, went to my car, and had a cry, because I'm a soppyarse.)
I don't know what your initiating airline is, or its process, but that was my experience. IAG's Philly office was full of very nice people, but they were disorganised and unable to explain various policies, which did not imbue confidence. The UK end was a different story — those companies set up to receive and process your pets seem to be full of employees who love what they do and who understand that some of us ::ahem:: are soppyarses. And everything turned out well in the end. Now just 11 days till snuggles!