What's the best way to estimate or determine the approximate number of systems doing mining and maintaining the integrity of the ledger (there's https://ethstats.net/, but that doesn't show all miners, of course). Also curious about ways to know the number of smart contracts deployed on the mainnet.
The best way to estimate the approximate number of systems doing mining is to use the hashrate or difficulty.
From https://etherscan.io/charts/hashrate, the current hashrate is 5,479 GH/s, or 5,479,000 MH/s. For comparison, my 4 x R9 390x produces 125 MH/s. So roughly 43,832 x my mining rig.
The current number of ethereum nodes maintaining the blockchain data is 6,228 - from http://www.ethernodes.org/network/1.
The number of smart contracts currently deployed on mainnet is 124,349. You can find the statistics from https://etherscan.io/accounts/c . An unknown proportion of these smart contracts would be the Wallet contracts.