Is there a good Desktop bitcoin app?

There is. See for a list here: https://bitcoin.org/en/choose-your-wallet
Best for me are Electrum and Multibit. Bitcoincore is good but it takes too much data for me.

Can I sync between my Android app wallet to a desktop app wallet?

You can. Import the privatekey in both of your android wallet ( suggestion for best android wallet to do this is mycelium) and import it in your desktop client. You can have access to yor wallet at the same time.

Are all wallet apps the same? or do some offer unique functionalities?

Yes. Some offers a more advance functionalities and some wallets offers basics.

Is my wallet unique to the app I used to create it? I mean, if I install a different Bitcoin app, would I be able to have my wallet appear on the new app?

Your wallet is tied with a privatekey or a seed which you can use in another app by importing it in the client.

Is it possible to share the wallet with my spouse? I mean, if something happens to me, I'd like her to be able to access the wallet and keep the Bitcoins

Yes. Give her/him a copy of the backup of your wallet. Incase something happens, with that backup (wallet.dat, privatekey or seed) s/he can access it.

If the answer to the above question is 'No', then what would be the best way to let her access the wallet?

It's possible. see answer above.

Let's say I (and all of us) became rich from Bitcoin investment, how could I change Bitcoin to dollars in my bank account?

You can use an exchanger. Most used and well known are coinbase, bitstamp, localbitcoin etc.

What is the best way to purchase Bitcoins? I'd like to start by purchasing 2 bitcoins. Is a purchase method unique to the country I live in, or is it done via the app I'm using?

There are bitcoin exchanges that operates only in certain county so yes, methods differ depending on which country you live but most exchanges , you can buy using a bank or a visa card , and there are also bitcoin atms where you can buy and sell btc.
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