What are taker and maker fees? Please see the attached pictures as a reference
If I sell my 0.9 Litecoin at 39.18 (assume that I can sell at this price), do I need to pay any fee?
What if I sell my 0.9 Litecoin at 39.15 (assume that I can sell at this price), do I need to pay any fee?
When a market charges taker and maker fees, they differentiate whether you're increasing the size of the order book or decreasing the size of the order book.
When you create an order that is immediately matched with already existing orders, you're a taker because you take liquidity from the market. When you add an order that doesn't match existing offers, you add liquidity to the market and are charged a maker fee.
For example given the following orderbook:

Alice sell 1 BTC at $1000

Bob  sell 1 BTC at $995

Charlie buy 1 BTC at $990

Dave buy 1 BTC at $985


If you add a buy order at $1000 for 2 BTC, you will immediately match the offers of Alice and Bob. You're a taker.
If you add a buy order at $993 it doesn't get matched and you're a maker.

In your case, both suggested prices for your sell order are higher than any buy offers, so you'd only pay the maker fee of 0%.