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  1. Use always an open source wallet app software. That means, other people with coding knowledge could check the app code for possible leaks.
  2. I don't understand your question about "transacting offline". Bitcoin transaction need to be confirmed online (are many different ways to check that). If is an onchain tx or LN tx both will have to have at least a confirmation. The only real offline transaction could be using OpenDimes, physical devices where keys of a bitcoin address are stored and even those have a very good mechanism to check if the keys were used or not.
  3. Even if your wallet app is fishy, you can check any tx you've done on any block explorer. Are many, for example: https://mempool.space or https://blockstream.info and see if is validated or not.
All information required to know how to use a BTC/LN wallet is there.