Do I have any reason to worry if my transaction doesn't confirm in a long time, I don't know days, or even weeks? Because during that time, I feel like my BTC is in "limbo", neither here nor there, and that's quite unnerving. Or is being patient the only requirement, and the transaction WILL DEFINITELY eventually confirm, and it's not possible for the BTC sent to "vanish", never reaching the receiving address?

Bitcoin's can't disappear. As long as your wallet keeps broadcasting the transaction, it'll confirm eventually. However, there's no "DEFINITELY" in this: as long as
fees stay high because users keep making more transactions than fit the blocks, low-fee transactions will be ignored (Bitcoin miners choose the transactions that earn them most profit).
Most wallets though stop broadcasting after a while. In that case the unconfirmed transaction disappears from mempool, and your funds show up in your own wallet again.
Whatever you do: don't start messing around with Forkcoins before your Bitcoins are safely confirmed in your wallet.