Hi, I have flown multiple times international (10+ trips) and domestic (5+ trips) with our Havanese. Here are my experiences and recommendations!
We fly with a Sherpa Delta bag (as we always fly Delta or codeshare). It is the same thing as a Medium Sherpa Deluxe bag, with the exception that it says "Delta" on one side. The best part about this bag is that if you are denied boarding a flight due to problems with the bag (attendants say it is too big, etc) you will be reimbursed for your plane ticket because of a partnership Sherpa has with the airlines. I believe this bag also comes as an American Airlines bag, so that could be good for any AA (or codeshare partner) travellers.
Security is an awesome way to show off your dog's wonderful obedience skills! You tell your dog to sit and wait, and you walk through or stand and scan through security, then tell your dog to follow. Stand back and watch everyone be amazed. Or, they ask the dog to go through first, sometimes through a detector, sometimes not, and a waiting personnel will hold her while I am scanned or patted down. The first few flights our girl was so small that the airport personnel just cooed over her and passed her down the line. (It always makes me shake my head when airport personnel skip my dog around security, as their job is to secure everything coming through, but oh well.) It really isn't a huge deal, and even with my layers of coats, shoes, dog, laptop, ipad, liquids in a baggie, Sherpa bag, and "personal" item (see below), I am one of the fastest people through security. While you are waiting to reach the conveyer belt, take off your coats, pull out your liquids, unzip/lace your shoes, and take off that belt. Then, put your bag and your bagged dog on the conveyer belt with the top unzipped (if they will stay) and get everything else in a bin first. After you get your electronics out and check your pockets, take the dog out and go. Reverse the process for the other side, and that is how you fly through security! Just remember to not worry about the people behind you and take your time.
When I fly domestic, I usually take a backpack as my "personal" item, as I can put my purse and my laptop or iPad in it. I also put any boarding documents my pup needs in this bag, as well as a little bit of dog food. When I fly international, I seem to get away with using a 20" upright (the rolling kind that sits on top of the largest standard size rolling carryon bag; usually the smallest in a rolling set). This is EXTREMELY helpful, because then you can carefully place the Sherpa bag on top of the rolling bag, saving your shoulder and back. [Let me interject here and say, yes, your carryon pet does replace your carryon bag, but the logic I have is that as long as the dog can fit under the seat where the personal item goes, I can have a "personal" item that goes in the overhead. I don't try to take a full size carryon, just something with wheels that can carry my electronics, contents of my purse, and important documents, all things which fall under the category of a "personal" item]. Though the Sherpa bag has nice pockets on it, I try to keep everything in my "personal" bag so that my pup has as much space as possible in her bag-no walls pushing in on her from being stuffed with supplies. The only exceptions are a bone to chew on, a collapsible water dish that stays tucked in the side, and potty pads that go under her cushion in the bag. We do not hook our girl in our bag, as she is really good about staying put. She only wears her collar with her ID tags, vaccination tags, and microchip info. We don't want her to feel any more restricted than she already is being stuck in a bag.
As far as pottying goes, TSA and airport rules say that once you enter an airport, all non-service dogs must be locked away until you exit your destination airport. Seeing as I offer as much water as possible to my dog due to the dehydration that occurs during flying, (plus about half her normal amount of food for the usual time period), pottying is something that likely needs to happen. We of course take advantage of designated doggy areas and use long layovers to leave the airport security and find an outdoor place, but if you can't leave the airport (customs, weather, close connections, etc.) I have found that airport bathrooms are a good place. I try to find an end area with little traffic where I can keep her on a leash and lay down a potty pad and let my girl out of her bag. If I can't find a quiet corner, I go into a stall. If it is a REALLY quiet bathroom, I let her wander around a bit. Never have I been questioned or lectured about taking my dog out of the bag in the bathroom. If I am ever bothered in the future, I plan to say "I'm sorry, (you don't offer a dog area so) I thought the restroom would be a proper area for her to toilet," Or, the good-old act dumb and say "Oops, I'm sorry, I didn't know!"
During international flights, I take my dog in her Sherpa bag to the largest lavatory I can find, put her bag on the closed toilet seat, shimmy around, pick her up while I delve under the pad and pull out a potty pad, put the dog back in the bag, lay out the potty pad on the floor so that she doesn't touch a single centimeter of the nasty space, place her on the floor, sit on the closed toilet seat with the bag in my lap, and tell her to go potty. Obviously, I do not bother with this during shorter domestic flights if I can avoid it, as the smaller domestic planes usually have smaller bathrooms, making this a rather tedious process. My girl seems to treat air travel like nighttime, where she doesn't use the restroom for 10+ hours, as she chooses to hold it in. We use the bathroom as an opportunity to stretch our legs.
Because of our lifestyle, we literally started flying with our Havanese as soon as she was old enough (12 weeks, I believe, on Delta). To prepare for this, we would ride around in the car with her in her Sherpa bag, and even would put the bag on top of the washer and dryer while they were on (probably a bit of an overkill, but we wanted her to experience noisy rumbling and shaking). She never seemed too concerned, but perhaps this is because we introduced it so early. We NEVER tranquilize, due to the dangers of it mixed with air travel, and based on the idea that it isn't the best setup for proper behavior when we, as humans, are heavily sedated and out in public. I was concerned about our girl barking or whining on her first flight, but she literally fell asleep as soon as the engine turned on. Additionally, the white noise on flights drowns out a lot of sound, and the frequency of it might just put your dog to sleep! Our first flight we put a rag that we had slept with for a couple nights in her bag just as a little extra security blanket.
Make sure you have your paperwork: very recent vet certificate, ID, passports, microchip certificate, rabies certificate, shot records. If you can get a passport for your dog, it makes things much easier. Additionally, the pre-travel vet check turnaround time and cost goes WAY down ($180 vs. $40) due to not needing a notarized certificate. I also print off copies of ALL airline and international travel policies according to the governing body of the area (European Union for most of Europe, USDA for the USA). That way, if something is contradicted to what I had read and prepared for, I can show them the paperwork that will help convince them.
Also, make sure that you have enough food and medicine for a day or two (depending on your expected travel time) due to unexpected delays. If you travel internationally, you may have some issues with needing to declare animal-based products (usually I only have trouble coming back into the US), i.e. dog food. One trip, even with the paperwork that said I could bring a small quantity of dog food with me back into the USA, all of my dog's food was confiscated. She usually chooses not to eat while travelling, so after a messy 27-hour journey with multiple delays that got us in to our destination at 11 PM, she was S.O.L. when all of her hypoallergenic sensitive food was seized. I found the best way around this is to stash about 3 of those snack-sized Ziploc baggies in different areas of my luggage. Am I risking biological contamination? Perhaps, but I will be ****ed if my dog doesn't get to eat for 36+ hours because someone disregards the rules that are posted on the appropriate governing body's .gov site.
I do believe that it is important to follow the rules and be prepared so that travel goes as smoothly as possible. But, all that being said, I do break a few rules for the overall comfort of our trip. Already I told you I take a rolling bag that is a little larger than a personal item would be so that I can place the dog on the wheeled bag. I also said I take the dog out in the restrooms, and that I hide extra dog food, just in case. Because our girl is excellent about staying in her bag, we unzip part of it so that her head can pop out when we are in the airport (but not on the plane! Even we do not risk the miniscule chance that she would jump out and run around, thus causing us to be de-boarded), and while we are waiting at the gate, I unzip the top of the bag so she has the option to stretch up and look out.
I think this is everything; I guess my whole thought on the process is to not be stressed about it. Remind yourself that though it may be stressful, the stress would be 10x as much for both you and your pet if they were to travel in cargo. You have so much more control about how your pet is treated, when they can potty, when you feed and water. And even when stuff happens that is outside of your control, the best way to deal with it is to be happy that you have your fluffy friend with you on this adventure called life!
Best of luck and happy travels!