Is it possible for a miner (say for example, I had a single connection to one miner) to accept a transaction fee for processing my transaction and then at the same time reject my transaction or not process it at all?
Not possible. Transactions are all-or-nothing. For transactions that include fees, the inputs are larger than the outputs, and the discrepancy becomes an input to the coinbase transaction and goes to the miner. Miner includes transaction in block, miner gets fee. No transaction, no fee.