If you have a block, where all fees are below 1000 satoshis, then it means, that you basically have less than 1 kvB of data to confirm. Which simply means, that nobody is using the network at all. And then, it doesn't make any sense to run any kind of lottery, if nobody would receive those coins.

Also, another reason for the minimal amount, is to allow the winner to spend it. Imagine winning 110 satoshis. A miner could obviously send it, but how the user is supposed to make a non-standard transaction, when that user is not mining?

Another thing is that nobody stops you from counting 110 satoshis as a winning amount. But in this case, it should be batched, and for example if someone will win 10 times, then that player can receive 1100 satoshis once, instead of receiving dust amounts in every round.

Ideally, coins from the lottery would be accumulated on a single UTXO, and users would know, how to withdraw the amount they won, at any time. But we are not there yet, because the main network does not support batching things on that level, and forming such coinpools in a decentralized way.

1. By default, every user is losing, and all fees goes to miners. Which means, that any non-zero amount of lottery nodes is better for users, than no such nodes.
2. As long as testnet4 is in its infancy, transactions could be free, or almost free, and it won't be harmful. So, that kind of lottery is just some kind of workaround for "free transactions during the early stage". Processing all transactions for free is hard, because even though there are some nodes doing that, then users are not really using that feature. So, it is a way to reward at least some of them, to imitate the initial stage, where early adopters could get their transactions processed for free. Ideally, it could be handled as 1-of-N multisig on all public keys, but it would be too spammy, so it is reduced to a "winner takes all" scheme.

If you think about alternative implementation of a magic OP_LOTTERY opcode, handling the whole thing, then it is still not resistant to being manipulated by miners. Given enough incentive, a miner can try to drop some block, and mine an alternative version, if it could give a bigger reward in a lottery. It is similar to reorging a block with 10k tBTC fee: if you have some transaction, with some payment to the winner, and the miner can check, if he is winning or not, then that miner can try to game the system, by trying different inputs, as long as it is profitable.

But yes, when selecting the winner, any reasonable input can be used. The bare minimum is the client seed (first pubkey), and the server seed (second pubkey), connected with some kind of nonce (for example as a block number). Then, by mining a block with a hash of some public key, it is an equivalent of revealing some server seed. Which means, that in any later block, a client can specify nonce in the locktime field, and reveal some public key, when spending the transaction.

If some transaction is standard, and it is broadcasted to the whole network, then any non-lottery node can also include it, even if some lottery node will exclude it. And then, a winning bet will still be considered, because after reaching a single confirmation, all nodes will count a valid winning bet, as a valid winning bet. The transaction locktime will give us the nonce, the client seed will be revealed as the public key, and there will always be a chance to get a match, if not in this lottery client, then another.

Also, for that reason, sharing commitments in a P2P way is important, because in this way, we can avoid a situation, where every key is rewarded by half of the lottery nodes.

One reason is that the centralized website, from which I linked the provably fair model, is getting worse and worse, so there is some room, to make a decentralized alternative. Another reason is that testnet4 is in its infancy, and users are unnecessarily paying 1 sat/vB, instead of getting free transactions, or paying 0.01 tBTC, as it was, during the early stages of the mainnet.

And of course, another reason is that I mined more than enough testnet4 coins, so I think about a good use case for them. Or rather: a use case for transaction fees, which reached something around 0.03 tBTC in my case, and it is growing. Because when dealing with coins before the first halving, their amounts are so huge, that limiting all tests into 0.01 tBTC (as the smallest unit) is good enough for everything.

Also, I thought about models like "collect 0.01 tBTC in fees, and then release it to a single winner, picked from everyone, who contributed into that pot". However, I wonder, if it wouldn't take too many blocks to get there, so people can get impatient, if they will have to wait that long, for claiming their reward.