diablo-3
Simple example – I'm in act II and found that liquid rainbow that could lead me to the Easter egg world. Since I found that item, I've forgot some that were in act I.
So, can I select a quest in act I to get those items and then select the last quest in act II to resume where I was without losing anything?
You will not lose the gear that you have gathered.
You will lose save progress if you start at part 1, you can go back to the quests you already done (In your example 7 to 9).
The way the party system works is that you always enter the game at the level that the host is playing at. If they are further ahead of your character personally you will join them at part 7 (in your example).
When you go back to your own personal game, you can start back at part 1 but it will prompt you to save over your spot in the game. You can continue on from part 9, but you lose out on the story line in between.
What you can do is have your friend join you in your game and help you kill the butcher and jump ahead after that is done.
Edit image: 
Here's what I've compiled, based on the Diablo lore and the Heroes of the Storm character backstories.
Female Demon Hunter: Valla
Diablo III: Hatred and Discipline, Demon Hunter short story
Valla went on, "It was a good dream... but for now it must remain just that—a dream."
The sawyer's daughter pulled her hood up. "I'm back. I'm back, and I'm ready... to continue the hunt."
She turned away.
"Where do you think you're going?" Josen asked flatly.
"Tristram. The others asked you to send the best. I am the best. I'm going, and you have less than a few heartbeats to try and stop me."
Valla waited with her back to the master hunter, then lifted her scarf... An instant later she strode away, crested a rise, and was gone from sight.
Heroes of the Storm: Valla, Demon Hunter
Valla witnessed the ferocity of the hellspawn firsthand as the demons ravaged her village and left her for dead. Now, with no people to call her own, her only allegiance is to her cause: to rid Sanctuary of the demonic filth corrupting its lands.
Male Witch Doctor: Nazeebo
Not named in Diablo III, however the character background in Heroes of the Storm is consistent with that of the heretical Witch Doctor encountered by Benu in the Doubtwalker short story.
You stray far from the truth. What is true is this: the Unformed Land is not as the high priests teach. The one you call heretic knew this. That is why he defied law.
Images swirled and flashed before Benu like smoke and lightning. He glimpsed the so-called heretic wandering through strange lands the young witch doctor did not know. A falling star blazed in the night sky, and Benu followed it to where it crashed on the earth—a small town beset by evil.
Heroes of the Storm: Nazeebo, Heretic Witch Doctor
With cabalistic incantations and vestments of feather and bone, the Witch Doctor Nazeebo calls upon spirits from the Unformed Land to aid him in battle. Few outlanders ever encounter a Witch Doctor in the flesh, without losing their own.
Female Barbarian: Sonya
Not named in Diablo III, however the character background in Heroes of the Storm seems consistent with that revealed in the game.
Heroes of the Storm: Sonya, Wandering Barbarian
From the frozen Dreadlands of the north come the mighty Barbarian warriors. After surviving the destruction of her home and death of her tribe, Sonya wanders Sanctuary in search of battle and a better life for her people.
Male Barbarian: Kehr Odwyll (questionable)
The peasant took a few quick steps ahead, trying to catch Kehr's eye. "I doubted at first. You lack the wild beard and locks mentioned in the tales..."
He cleared his throat.
"If you do not want to speak, I understand. I wanted only to thank you."
Tilting his head in a bow, he let the barbarian stride past.
Female Wizard: Li-Ming
Diablo III: Firefly, Wizard short story
But understand this: Li-Ming did not lie. There are tomes within our library that describe the events that may now be coming to pass. It all begins with a star falling from the heavens, and such a star did fall the day I fought Li-Ming.
I know the true nature of magic and of who and what I am. Li-Ming knows these things as well, but she has come to a different course. This is the puzzle put before us, assassin. I am not blind to the evil that stalks us, but I fear for what Li-Ming might seek to undertake. Thus I consign to death my brightest student, perhaps the world's best hope for salvation, and pray that I have chosen rightly.
Heroes of the Storm: Li-Ming, Rebellious Wizard
It’s a generally agreed upon fact of life among the denizens of Sanctuary that wizards and magical mayhem are never far apart. Although she understands people’s concerns, Li-Ming is entirely unapologetic for the destruction in her wake. She delights in unleashing devastating spells on her foes from afar, then teleporting away to frustrate her enemies’ attempts at striking back. Small minds may fear this kind of power, but Li-Ming wields the arcane forces of the universe with confidence and unmatched skill.
Best Answer
You can change quest, but you'll have do the quest from the beginning, instead of resuming from where you were. If you were at the beginning of a quest, you lose nothing. If you were in the middle of a quest, you may have to repeat a little. But I think it is not a big deal. You'll probably farm the game for hundreds of times anyway. :)