I know that generally Double substitution rule disables substitution chains so that for an example Rumble Lecter can make a Shield Force active Perfect Galaxy to leave the battle zone.

l'd still want to ask a confirmation for a few situations. Firstly, Glais Mejicula can replace shield breaking by discarding your hand cards. Does double substitution rule prevent you from putting cards like Arque Delacerna into the battle zone instead of discarding them (with Glais Mejicula) during your opponent's turn. Also, has double substitution rule always been applied, meaning that this rule wouldn't only apply for OCG format?

Secondly, can a duelist discard Arque Delacerna into the battle using Gigastand's ability: "When this creature would be put into your graveyard from the battle zone, you may return it to your hand instead. If you do so, discard a card from your hand". In this case, discarding could be interpreted as a separated case but it is still directly dependable on substituting Gigastand's placement to graveyard with returning it to its owner's hand and then discarding a card. Phrase "if you do" makes it more clear that you have to discard a hand card if you decide to return Gigastand to your hand instead of putting it into your graveyard.

These questions have risen in our local DM community. Thanks in advance for any help.

You should be able to get Arque Delacerna for free with Glais Mejicula according to wording 'discard' it seems to be correct to me. The only condition is that you have to discard Delacerna in your opponnents turn. I can't say how it works with the double subsitution rule, that's always a point worth to discuss.
I'm sure you already checked it. If not: http://duelmasters.wikia.com/wiki/Terradragon_Arque_Delacerna/Tips

No you cannot, there's a thing called the double substitution effect.

Thats a one f******* joke! Glais Mejicula and cards with that ability have created long before "double substitution". That rule is clearly for OCG where is multiple epic chaining chances. I'd say TCG format CAN CAN CAN CAN use Delacerna, Bingole, Sanfist, Dava Torey free spam with Glais Mejicula.

It would make sense to play TCG normally without the rules that came a long time later for the OCG, like Giant Insects also being considered Giants, and the new revision(http://duelmasters.wikia.com/wiki/March_20,_2017_Rules_revision). However, I'm not sure if the Double Substitution ruling came out officially at some point, or if it was present from the very beginning. Although most players accept that Mejicula CAN bring an Arque out.

As for Gigastand, you are correct, the discard is not a substitution but an additional effect, and the anti-discard ability can be freely used.

I second Zachs answer!
It pretty much explains what i would have said

nitrox wrote:As for Gigastand, you are correct, the discard is not a substitution but an additional effect, and the anti-discard ability can be freely used.

Gigastand could be seen as a double substitution related effects since, in other words, it could be understood similarly to the following: "When this creature would be put into your graveyard from the battle zone, you may return it to your hand and discard a card from your hand instead".Now you can clearly see that discarding is a substitutive effect for the destruction of Gigastand and thus you could link Arque Delacerna's substitutive effect for discarding with that due to the Chain Rule of Double Substitution rule. I think the only reason why the ability in Gigastand is "divided" in two parts is to make it more clear in which order to proceed.

nitrox wrote:However, I'm not sure if the Double Substitution ruling came out officially at some point, or if it was present from the very beginning. Although most players accept that Mejicula CAN bring an Arque out.

This is the main problem. I think that Double Substitution rule has been always around in a way or another. Nevertheless, In Finland we decided to ignore Double Substitution rule entirely since it doesn't seem to bring anything game breaking in the mix. If the situation changes, we will reconsider. In OCG, we naturally apply Double Substitution rule normally.