There is no such thing. There's no free lunch. Anyone who comes up with your features will charge you with ads or in-app purchase. Nobody will do this.
Android chess app recommendation

Thanks for your replies.
The app I'm looking for needs to work off line (not connected to the Internet) as we may not have an Internet connection while traveling. It should connect 2 Android phones via Bluetooth (or, less preferably, WiFi direct). Primarily, it should take the place of a travel chess set but having the flexibility to allow us to play without needing to face each other.
@student-t I'm hoping there is one. There are many good free software, even my phone's OS, Android (Linux variant), is free.
@Nathan0001 I think the chess.com app is very nice, but it needs to connect to its server on the Internet. We will not have this connection.

Android is free and open source, but it's done to promote Google's own advetisement. The intention is still to make money. Android app developers won't gain anything unless it shows ads or purchases.
You won't find anything that can satisify your very specified requirments. You should try multiple apps.

@student-t
@uri65
Thanks for your suggestions. Flipping the board each move just seem too inconvenient; plus, I don't see it working well for a 5-min blitz game. Multiple apps can be an option, where son and I can use the chess.com app when we have power nearby and wifi; but I still dislike ads.
Anyway, over the weekend, I spent some time writing an app for my needs. It doesn't seem that difficult. Definitely easier than my search so far.

I haven't tried this but a chess program, Chess by cnvcs.com, but I thought of your question here when I saw in a thread at http://www.chessreaders.com/2014/09/top-ten-chess-apps-for-android-devices.html that: "Best part of this app is you can play with other players via Bluetooth or WiFi."
is here:
https://play.google.com/store/apps/details?id=com.cnvcs.chess
and it claims:
Features:
- Play as white or black, against the computer or a friend.
- 10 difficulty levels, from novice to grandmaster.
- Include more than 38000 chess puzzles, divided into 13 collections.
- Support online gameplay.
- Support LAN game via bluetooth/Wi-Fi.
- Computer hint and unlimited undo for beginners.
- Changeable chessboards and piece sets, 2D/3D piece styles.
- Load and save games to PGN files.
- Autosaving your current game.
- Setup position, intelligent validity check.
- Randomly go to any point in the current game.
- Provide thousands of classic chess games to download, such as grandmaster games, world championship games.
- Switch to play mode when in view mode, analyze the game, and then restore the view status.
- More functions: handicap, view move list, automatic move playback, etc.
- It is free!
I'm going to download it myself and try it out and would do so before wrtiing this but since I have to start getting ready for a Super Bowl party soon, I figured I better write this in case I forget this thread later!

P.S. I downloaded the "Chess" by cnvcs.com app from Google Play to my antique Android 4.4 phone and somewhere in the process the words "contains ads" appeared, but I haven't seen any in my brief experience. I played a game against it's 5th highest of 8 levels and it played like a sub-1000 player. For playing I'd rather use Droidfish's Cuckoofish or Stockfish engines, limited by a percentage of ability. I do like the 38,000 puzzles in Chess by cnvcs! Chess takes up only 10.55 MB and Droidfish only 10.01 MB. On a phone with only about 2 GB of usable space, that sure beats the Candy Crush and Shoot-Em-Up type games I keep on the phone to pacify the young kids in our family.
I don't have a friend with Bluetooth or WiFi capabilities to test it against, but the initial menu includes Play, Online, and LAN. LAN brings up a menu for choosing Bluetooth or WiFi LAN. Bluetooth brings up a menu to Host Game or Join Game - I didn't go any further since that was pointless for me, but I saw no ad along the way. On Google Play, there's at least one complaint about online connections, but since I would be using the chess.com app for that, I didn't look into it.
I also checked out cnvcs.com. It's a Chinese website devoted to game apps (Chinese Chess, Go, Chess, etc.). A search turned up no warnings about it.

@MickinMD Thanks for the posts and detailed review.
I actually saw this app before making this thread and rejected it because I didn't see time controls. But after reading your initial post, I decided to downloaded it anyway. It is a nice app, but it doesn't have time controls (as I had originally thought) so it doesn't meet our needs. (We've been playing mostly 5- and 10-minute games ever since we bought our chess clock and really enjoy the added excitement.)

Anyway, I can't believe that it's been 3 weeks already; the first week made good progress, then 0 progress until yesterday and today : ) That's actually very normal for me. Here's my theory: there's never enough time for anything until it's really needed. (And I need this App tomorrow)
It's still very rough but meets our primary needs of blitz games and PGN records. I like the chessmen designs, but my son hates them -- so I'll probably redesign them.
Feel free w/ questions/suggestion (will be offline for about a week), and I'll try to respond.
Totally free, I'm not making anything off any of my Apps. They are mainly written specifically for my own needs or at the request of my son. If you find them useful, I'm glad.
https://play.google.com/store/apps/details?id=com.adamtai.chessfor2


Is there an app like this for ios?
yes.
What is the name of the app?
There's the chess.com app for one. But there are tons more.
Anyway, I can't believe that it's been 3 weeks already; the first week made good progress, then 0 progress until yesterday and today : ) That's actually very normal for me. Here's my theory: there's never enough time for anything until it's really needed. (And I need this App tomorrow)
It's still very rough but meets our primary needs of blitz games and PGN records. I like the chessmen designs, but my son hates them -- so I'll probably redesign them.
Feel free w/ questions/suggestion (will be offline for about a week), and I'll try to respond.
Totally free, I'm not making anything off any of my Apps. They are mainly written specifically for my own needs or at the request of my son. If you find them useful, I'm glad.
https://play.google.com/store/apps/details?id=com.adamtai.chessfor2
I checked this out and was super impressed with the way it works. As you say some design improvements would be great to see, overall meets need.
I wanted to be able to play with my son back seat and my front seat in the car without passing the phone was great.
Thanks. If you decide to open source the code, we could work on it together.
I'm looking for an Android app that lets me play chess with my son on 2 phones, e.g. when we are traveling on the airplane. The first few I saw/tried didn't meet my needs, and I'm hoping you experts would know something that might fit better.
Here's what I would like to have:
1. Like the chess.com live chess. We want to play on 2 phones and prefer Bluetooth networking (for its low power requirement as games can take a while and phones may have be charging).
2. Blitz mode or basic time controls. Like chess.com live chess.
3. Annotated history of played games on each phone. Like chess.com live chess.
4. chat ability, like chess.com. I'd like to share thought behind moves with him during non-blitz games to improve our games.
5. Free download, no ads, no in-app purchase. Since my son is only 6, I'd like to avoid exposing to too much ads and definitely no in-app purchasing.
6. Chess move recognition database, where it can tell me the opening names; again like chess.com.
Thanks!