Does the transaction fee of bitcored or bitcoind change everyday?

It will often change because it will estimate the fee for certain confirmation targets (e.g. confirm within 4 blocks). The fee estimation will update with each block, so it may change every day. It may also change more or less frequently than daily depending on the transaction situation for each block.

Am I required to set transaction fee of bitcoind by using settxfee method?

No. You only need to do that if you want to specify the transaction fee instead of using the estimation. I recommend that you use the estimation as setting the fee yourself will often occur in overpaying or underpaying.