The flowering stems are called scapes and are tasty when young and tender. Yes it is best to remove them soon after you see them – only cut out the stem which has a flower bud and not any leaves – to encourage bulb formation, but on elephant garlic this is not recommended, I am unsure why.
Most softneck garlic (Solent Wight, Thermidrome etc, and I think Sprint?) does not produce scapes.
All bulbs can be harvested ;ate June to early July, if you want best-sized bulbs. Harvesting later means they store less well as skin is shed in damp soil. Harvesting earlier means you have a fat, tender stem of garlic with little bulb. I hope this helps.