Do we need to buy the Cinque Terre Card for it
Yes. You need to purchase the card to hike any open segment of the Sentiero Azzurro. You can buy it at the train stations, Welcome Centers or at ticket checkpoints at entrances to the path. Doesn't matter if every one of them are inspected or not, fees from the card go towards (constant) maintenance of these overused trails so it's a must-do.
How much is the entrance fee (only) to the Blue Path?
As Frank stated, 1 day, adults: € 7,50, unless your accommodation offers the voucher for the discount.
http://www.parconazionale5terre.it/page.php?id=157
4 - 5 pm on a Tuesday in June. Has the crowd and heat subside by that
time?
I would expect it to still be plenty warm and busy enough. This path should not be hiked in or after heavy rain, and they will shut it down if conditions are unsafe to do so.