I shipped Logan home when I bought him.
DO NOT sedate her. Sedating affects regulation of body temperature, and it's too easy for her to get overheated or too cold if she's sedated. There's also the possibility that the decrease in pressure could affect how the drugs work. Best thing you can do to acclimate her to travel is to make she she's VERY comfortable in her crate (the crate she'll ship in, or at least the same style) and make sure she's comfortable riding in the car in the crate. It's not the same as flying, but it's about as close as you can get. Logan was super relaxed when he flew - I saw him on the tarmac when they were moving him to the next plane, and he was just chilling in his crate as they were moving him.
I flew Frontier, and I highly recommend it. They send a note to you when your dog is on board, which REALLY helps with nerves. I think there are other airlines that do this now, too. Frontier has climate controlled and pressurized cargo areas (and I'd guess most, if not all, airlines have this too), so the reason there are limit in place for temperature is for when your dog is on the tarmac getting ready to be loaded or after they're unloaded.
I think Logan cost an addition $100 on top of my ticket to fly one-way, which is a really good deal since it costs $250+ to ship them alone. There are limits to how many dogs can be on the plane, so book early to make sure you get a space. I booked online and then called immediately after to get a space for Logan. Then I called again a day or tow before I left to make sure they still had me down for a dog on the return flight.
Make sure the crate you get is big enough for her to stand up naturally in, INCLUDING her ears. Tango's crate was HUUUGEEEE when I shipped him becasue he is tall and has big ears. I would have felt comfortable sending him in a crate a size or two smaller, but airline regulations forced me to go up in size. Logan's crate was actually a tad too small, so we just made him lay down when they checked :rofl1: It was plenty big enough for him, but his ears were questionable (thankfully they were also glued down lol). Some airlines are more picky than others, and I'm sure some individual people are more picky than others. But I'd rather be on the safe side. That said, some planes can only take crates under a certain size, so it's best to try to work with your airline on finding a crate that's large enough and still fits in the plane.
As far as what kind of crate, I bought a Remington crate sight unseen and had it shipped to Logan's breeder. When we opened it...we were less than impressed. I really, REALLY like the crate for shows because it's super east to put together and take apart, but it's not very solid or secure. Ideally it would need to be reinforced at all the fasteners with zip ties. So find one that has screws/bolts to hold the two halves together instead of clips/pins. Make sure there is plenty of ventilation. Vari-kennel is the standard, but I've seen crates for much less that look more solid and have more ventilation. Whatever you get, make sure it says it's airline-approved, and if possible, go look at it yourself before you buy it. Then call the airline and ask them to double check that it's the right crate.
You need to have something absorbent in the crate, as well as a bowl for food and water. I used a towel in Logan's crate, but his breeder pointed out bath mats are EXCELLENT for crates, as they don't slip. So use a bath mat
Put the water bowl in the freezer a day or two before so you can put the bowl of ice in the crate. This means less spilling, and your dog won't gorge themselves on water and vomit. They make you put a bag of food on the crate in case something happens, too, though they're not supposed to feed it to your dog unless you tell them to or something happens. BUT, when I picked up Logan, there was 1 kibble in the bag, so apparently they either fed it to him or they spilled it. But if they did feed it, he ate it all, so he was relaxed enough to eat lol. You also need "LIVE ANIMAL" stickers on the crate, and they SHOULD have those or you at the desk when you check in. I looked EVERYWHERE for those stupid stickers before I shipped Logan, and after I finally found them, I found out there was also a set in the box with the crate AND they gave me a set at check in. They'll help you fill out the sticker when you get to the desk.
After they inspect your dog's crate, zip the doors shut with zip ties. There is no reason for them to open the crate door (so long as you mount the water bowl on the crate door so they can fill it from outside if needed), and I feel the zip ties deter them from opening it, too.
I've never shipped a cat, but I would imagine it would be the same as a small dog. In-cabin shipped is different than cargo shipping (but costs the same, so expect to pay $100ish per cat as well). The thing to check with them is to make sure you can bring the cats AND Em in the cabin. Since Em's supposed to go in your footspace/under the seat, and the cats are supposed to go under the seat, I can see them not allowing them together simply because there's no room. So double check that. As for the heart murmur, that is up to your vet. You need a health certificate no more than 20 days prior to flying, so if your vet feels like it's not safe for your cat to fly with a heart murmur, there's not much you can do about it. As for being shy, it should be okay - Tango was super shy about people he doesn't know, and he was fine. Another reason to ziptie the doors shut.