a. I'll be running a full node ?
Yes. You won't necessarily be acting in the same capacity as other nodes though. At first, you'll only have outgoing connections, meaning your node is reaching out to others for its data. You generally must port-forward port 18080 in your router in order to enable incoming connections, where you would then be feeding blockchain and mempool data to other nodes.
b. Will it be actively mining ?
Negative. It will not automatically start to mine, but you can mine. Mining is a function of either the wallet or a mining software. If you would like to solo-mine, once the blockchain is synced, and the wallet has fully refreshed, you can tell the wallet to start mining.
However, most people who mine will mine on a dedicated machine, using mining software, and they mine to a pool, since a pool has a greater collective hash rate and therefore a better overall chance of winning a block. You don't need to have a local copy of the blockchain to pool-mine.
c. If it will be mining - can I choose which transactions to pickup for mining ?
The transactions chosen will automatically be the most profitable ones. You likely won't have to "worry" about this, however, since you're very, very unlikely to ever solo-mine a block at this point in the Monero ecosystem's evolution.