There is nothing like that in the actual rules, you do not need to throw anything to summon the pact weapon. It is just as you yourself quote, if you do it in a battle it costs an action (so you may want to keep a summoned pact weapon in a scabbard always).
Most DMs would be inclined to house-rule the other way, making summoning easier (a bonus action or an object interaction), because it’s Cool to summon the weapon in the battle, instead of drawing it from a scabbard lika an ordinary weapon. Anyway, if your DM insists on the summoning being unreliable, just keep the weapon summoned and in a scabbard.
About Hexblade patron and Pact of the Blade pact boon: these are separate. You can pick Pact of the Blade with any patron, and you can pick any boon with Hexblade. They are separate features with very little interacation (Hex Warrior allows you to use CHA with any summoned weapon). The major defining feature of Hexblade is the Hex, not so much the blade.
Then you seem to imply you are testing against your own spell DC? If that’s indeed what you do, well, that’s kinda… to be frank, it’s plain stupid, as the DC becomes harder as you level up (proficiency bonus). Spell DC is what others roll when they try to resist your spells (and only those which call for a saving throw). But maybe you didn’t mean this.
Also, spellcasting DC does not interact with Eldritch Blast in any way. With EB you do a ranged spell attack (for Warlock: d20 + proficiency bonus + CHA bonus), and see if you hit the Armor Class of the target. Just read the spell description, and read the combat rules in the PHB (there are ambiguous bits in the rules, sure, but this is not one of them).