If I sent 0.002 or 0.005 from wallet to binance for exchange, I only pay a transaction fees from my wallet while the exchange platform (binance)didn't charge me anything.
You want the platform to charge fee for deposit? Is their any exchange that charge for deposting coin? I have not used such an exchange before.
So I don't know where they are getting their profit and pay workers.
Binance get money from many means like trading fee, withdrawal fee (which is far more than the transaction fee), ads display for p2p, IEO and mnay jther ways.
No. Noncustodial wallet do not charged from the transaction fee, the transaction fee is an incentive received by only miners, not wallet developers. You need to read what others poster like OmegaStarScream and ETFbitcoin posted.