Transaction vaildation
First of, each node on the network maintains an UTXO-set. Not only miners.

to 1. The UTXO-set is created by applying all transactions in each block in the chain from the beginning. All transactions that remain unspent when reaching the chain tip are in the UTXO-set. The older a block, the higher the probability that the whole network has a equal block. Equal blocks result in an equal UTXO-set.

to 2. Each node validates blocks, not only miners. A block is deemed valid, if it follows the consensus rules. One consensus rule is, that each spend transaction (TXO) must be in the current (pre-acceptance) UTXO-set or a TXO created early in the same block. Block validation does not interfere with the actual mining (only for a short time, when the block is accepted and a new blocktemplate has to be pushed to the ASIC). Validation happens on the host CPU and the actual mining part happens in the ASIC. If a miner doesn't validate an invalid block, he wastes his time/electricity/money with mining an invalid block on top of the previous invalid block.