First Time on Yugioh, Question...

Yesterday, I was looking up the Yugioh wiki for rulings on something called "Priority."(First time on Yugioh) At first I thought I fully understand it, until a watch several duel, and it made me more confuse then before.

1(First duel I watch)
The first duel I watch was "Player A" set a monster face down and set one card in his S/T zone. "Player B" summon a monster with 3000 atk point(Which happen to be Black Luster Soldier Envoy of the Begging), but "Player B" activate his face down card which is known as Bottomless Trap Hole to remove Player A monster from play. However, "Player A" called priority due to his turn to ignite his effect by removing Player B monster from play resulting a chain link. Both player agrees. His monster die and the opponent monster also die.

(The monster Black Luster Soldier have an ignition effect)

At first I understand it very much until...I seen this due.

2(Duel I watch)
So, "Player A" which was his turn at the moment flip a monster face up. A flip effect monster known as Man Eater Bug to destroy one monster on the field, but "Player B" activated his trap card "Solemn Warning" to negate the monster summoning. "Player A" say he call priority because of his turn, but the other player disagree and stated the monster were negate and destroy before it were summoned meaning "Player A" effect never happen/resolve. Because of this they called an Admin to settle the problem. The admin explain there's no such thing as "priority" and Player A monster is destroy before it even was flip summon.

(The monster Man Eater Bug is a trigger effect)

Now, What I wanted to know which one these elaborate and make more sense and if you guys could clarify more then it could make my life much easier. I just started Yugioh because of my friends invited me into it. And have read all the rules and this the only one that giving me a headache. I also apologize for the grammar, since I'm typing while having headache making me hard to focus on this matter. Is there really a thing "Priority" are it just made up? I search it on google but, starting to have doubt. Thanks in advance for your effort in helping me. And also can you synchro summon in a middle of a duel and still attack;considering it special summon?

Thank you.

Example A is a correct use of Turn Priority to use an effect.

Example B is someone not knowing what they were talking about. Solemn Warning negated the summon of the Man-Eater Bug. Its effect would not have the chance to activate because it was not properly flip summoned.

You can read this link for more information :

http://yugioh.wikia.com/wiki/Priority

The 1st situation is good. Howver the correct way of handling it would have been Palyer B asking Playe A if he uses his priority or pass it. If he uses it, BLS will banish the monster but Bottomless can be chained resulting in BLS being removed from play.

As for the 2nd situation, Palyer A has no priority since Solemn negates the summon, which means Man-Eater Bug wasn't properly summoned so he can't activate it's eff. Cards like Bottomless doesn't negate the summon.

Thanks you guys so much! It help me a lot on this matter.

There's a sticky that go in depth of priority
Watching the duel isn't the best way to learn since there are a lot of miscommunication or jumping the gun

And also can you synchro summon in a middle of a duel and still attack;considering it special summon?

Yes, as long as you synchro summon in your main pahse 1 theres nothing that doesnt lets you attack. Specials summons doesnt do that.