I have mostly stayed in private rentals in my travels, since long before Air B&B existed. I guess I'd say that of all the rentals I've had, the one that was booked through Air B&B was the worst, although it was by no means bad, and, to be fair, some of the rentals I've booked through other sites also advertise on Air B&B. I know other people who have used it and had great experiences.
One thing I've noticed is that sometimes an automated message in the Air B&B reviews will say that a rental has been cancelled by the owner within xx days of the stay, and that makes me a bit leery. I don't want to be scrambling for accommodation right before a major trip. I did have a booking that I'd made through Air B&B cancelled, but in that case, the owner cancelled in plenty of time for me to find an alternative, so it wasn't a big issue. But, for that reason, my preference is to book through other sites, such as Trip Advisor or Expedia, that offer guarantees on some bookings.
Another disadvantage of private bookings is that they don't have 24-hour front desks. If your flight is at an odd time or gets seriously delayed, this can be challenging if you have to meet up with someone to get the key (not always the case--sometimes there is a key in a coded lock box, for instance). The people I've dealt with have been understanding about this, but I feel bad when it happens. There also may be nobody to ask for directions or to leave your luggage after checkout.
That said, this type of accommodation can offer you more space and different amenities than a hotel. For instance, you may have cooking facilities if you want to prepare meals and save money, or at least a fridge and microwave for snacks, cold drinks, and reheating leftovers. You may have a washing machine. You get more of a "living like a local" experience. And the hosts are often extra helpful in providing information about navigating their city and will recommend sights and restaurants. Often, the host will leave a little welcome package to get you started (milk, wine, etc.). And, all of this typically costs less than a hotel room, with lower or no taxes.
For stays of 3 days or more, I usually try to book a private rental or an apartment-style hotel. For stays of less than that, I will book a hotel or B&B or inn.
Edited to add that I have never brought a gift.