What happens an order and offer are matched but there is a significant gap between them? I wonder the implications of the following scenario.

Let's assume I offered one coin with the price 1k (by mistake or for experiment), while the highest order in the market is 2k. What are the prices I'm going to pay and the matching order is going to receive. How would this affect the current price of the market? Thanks
It depends on the exchange; they will have rules for how order matching is handled.

On at least one of the exchanges I have used, if there existed a buy order for 1 coin for $2000 and you added a sell order with a $1000 limit, then you would get $2000. If however, the buy order at $2000 was only for 0.6 coins then you would sell 0.6 coins at $2000 and the remainder of your sell order (0.4 coins) would be matched by the next highest buy order, eg $1950, provided that another buy order existed at or over $1000.

I would expect this behaviour to be more common amongst exchanges that what @chytrik described in his answer.