Usually it is their own transactions that are broadcasted together with blocks found by them.

You can check with the filter and some parameters from Blockchair.com explorer.
https://blockchair.com/bitcoin/transactions?s=fee_per_kb_usd(asc)#f=block_id,input_count,time,fee_per_kwu,fee_per_kwu_usd,fee_per_kb,fee_per_kb_usd

If you're a normal Bitcoin user, consider 1 sat/vbyte is a minimal fee rate you can use for your transaction. Bitcoin nodes from miners can have their own setups to exclude transactions with fee rates lower than a number like 1 sat/vbyte or whatever.

Perhaps transactions with fee below 1 sat/vbyte can be accepted but that's basically the choice of the miners whether to accept or reject it but its practically possible.

Minimum miners are paying these days is 1 sat/vB, but we usually see in mempool.space is something around 5 sat/vB, so I would not risk going with bear minimum.
Back in the days it would be possible to sent bitcoin transaction without any fees, but mining today is more about profit.
You can however always start your own mining operations even with solo miner, but don't expect it to be very profitable.

Unless you get lucky and be included in 0 sat/vB transactions



But I far as I know the minimum is 1 sat/vB.

I also doubt that there are miners who changes the default ^{minrelaytxfee} line to bitcoin.conf.

Probably meaningless if the propagation is poor, ie. if majority of the network doesn’t lower their mintxrelayfee Miners probably has a custom implementation of a Bitcoin node so it wouldn’t be out of the realm of possibility that they would include these in their mempool (miners regularly mine nonstandard transactions anyways).

 I’m inclined to think that certain miners would operate in this manner. However, its probably difficult to tell given that mempools are often filled with txes paying above the minrelayfee.