LOT is the Polish airline. It codeshares with JetBlue, which was convenient because JetBlue had good choices from MCO (our home airport) to JFK.
I used to fly back to Hungary every year. I spent many years cramped in coach. This was fine when I was young, but now I am old and do not like the cattle call of coach on a 9 hour flight. For me and my husband, we fly business class now. I stick my kids (they are adults!) in coach (they can always pay the difference to business class, but they don't). They are still limber enough to deal with it.
If you find flights you like, check out the credit card for the airline. My AA card gives me a piece of checked luggage with no charge. This works well for domestic coach flights. If your preferred airline has a card with luggage included, and you would rather stay in coach, this might be a good option when it comes to including luggage.
The other thing to look for is that some airlines include a piece of checked baggage for an overseas flight, whether you fly coach or higher.
To save money, you can check breaking up flights, and you don't have to fly a codeshare connection. Codeshares are definitely more convenient though, because then your checked luggage will greet you at the Budapest airport, and you won't be rushing to pick it up and check it through after your first leg. However, if flying, say, Southwest to Chicago, then AA to Budapest, saves you whatever XX amount of dollars per person, and you are comfortable with getting luggage and then rechecking it, that's your call. I have flown to Europe many times with many different connections, and for me, it isn't worth the hassle, but I like things easy and not rushed. You know your situation better than I would.
Lodging - are you staying in Budapest or visiting relatives in one of the other cities in Hungary?
If in Budapest, are you visiting relatives there? Or are you just going to see the sights and immerse yourselves in all that is Hungarian? This will make a difference on where I recommend staying.
There are 2 sides to Budapest - Buda, the hilly, green, quiet side, and then across the Danube is Pest, the busy city. To go to either side, you have to cross one of the bridges. A car is not necessary, as public transportation is phenomenal! In fact, driving in the city is something I don't do, because it is so busy. The city was not built for all the cars that exist there now. Public transportation is the best option.
I would suggest you get a guidebook and look at the various sights, then decide what you would like to see during your stay. It would be easier to make a decision on where to search for an AirBNB then. However, I get wanting to see the options just to have an idea, so for searching, assuming you are visiting city sights, search for anywhere in the V, VI, or VII districts. Then read reviews.
Times have changed, and it used to be that the only places you had to avoid are the gypsy areas. Now, gangs from Russia and China have entered into some of the neighbourhoods, and drugs have arrived, so things are different. On the plus side, they aren't in the tourist areas, and the Russian and Chinese mobsters tend to just attack each other, but the drugs they bring in sadly contributes to the homelessness in Budapest. Also, the police are just learning about personal rights, so they are very quick to grab criminals and get them off the streets without worrying about the rights of the criminals (and if a crime happens, they have no qualms about shutting down a whole neighbourhood until the perpetrator is found). It is a bit different there.
Honestly, I feel safer walking Budapest at night than I do in some other areas here in America. And, Hungarians are so shocked when crime happens, that it is all we talk about, quite sure that the sky is falling now, so when 2 murders happen in one year in a town of 250,000+, we are convinced lawlessness is rampant and we are doomed. In reality, crime is there, but as a tourist, if you experience crime it will most likely just be a pickpocket, but it you are smart and aware you should be fine.
I do not recommend going to the bars and leaving at 3 in the morning drunk out of your mind. Then, you will probably be mugged. It happens quite often to the Brits that come over to Budapest for Stag and Hen parties. Drunks people flashing cash is never a good thing.
Also, make sure that wherever you stay, there is air conditioning. Hungarians have not embraced the a/c in their homes as we have in America. It can feel very hot in the city, hotter than in the countryside, so a/c is a must if you don't want to sweat.
Also check for elevators. And, the 1st floor in Hungary is actually identified as the "ground floor". The first floor would be what in America is classified as the "second floor".
Another thing to keep in mind - check the sizes of the beds. I like my king sized bed here in America. In Hungary, you may get a full sized bed for 2. Heck, in Hungary a twin works for 2 people, also. Personal space is not that important, and they just don't have the space for the large beds we are accustomed to. So, even if the AirBNB says it sleeps 7, check to see HOW it sleeps 7! Two twins, and a bunk bed, just would not work for my family!
Sorry, I know this is quite long. Hopefully I was able to answer a couple of your questions though. Let me know if you need any clarification!
