What are Lutheran teachings on salutary effects of communion?

Perhaps what you are looking for can be found in the Lutheran Small Catechism.

What is the benefit of such eating and drinking?

That is shown us in these words: Given, and shed for you, for the remission of sins; namely, that in the Sacrament forgiveness of sins, life, and salvation are given us through these words. For where there is forgiveness of sins, there is also life and salvation.

How can bodily eating and drinking do such great things?

It is not the eating and drinking, indeed, that does them, but the words which stand here, namely: Given, and shed for you, for the remission of sins. Which words are, beside the bodily eating and drinking, as the chief thing in the Sacrament; and he that believes these words has what they say and express, namely, the forgiveness of sins.

You should check into the Book of Concord as there are several sections (of which the Small/Large Catechism is a part) that address this specifically (http://bookofconcord.org/sd-supper.php - The Solid Declaration of the Formula of Concord).

"Did" is also implied to a degree, because without Christ having done his work, the sacrament wouldn't exist.

I'll extract some sections that pertain to your question:

[61] There is, therefore, a two-fold eating of the flesh of Christ, one spiritual, of which Christ treats especially John 6:54, which occurs in no other way than with the Spirit and faith, in the preaching and meditation of the Gospel, as well as in the Lord's Supper, and by itself is useful and salutary, and necessary at all times for salvation to all Christians; without which spiritual participation also the sacramental or oral eating in the Supper is not only not salutary, but even injurious and damning [a cause of condemnation].

[62] But this spiritual eating is nothing else than faith, namely, to hear God's Word (wherein Christ, true God and man, is presented to us, together with all benefits which He has purchased for us by His flesh given into death for us, and by His blood shed for us, namely, God's grace, the forgiveness of sins, righteousness, and eternal life), to receive it with faith and appropriate it to ourselves, and in all troubles and temptations firmly to rely, with sure confidence and trust, and to abide in the consolation that we have a gracious God, and eternal salvation on account of the Lord Jesus Christ. [He who hears these things related from the Word of God, and in faith receives and applies; them to himself, and relies entirely upon this consolation (that we have God reconciled and life eternal on account of the Mediator, Jesus Christ),-he, I say, who with true confidence rests in the Word of the Gospel in all troubles and temptations, spiritually eats the body of Christ and drinks His blood.]

[63] The other eating of the body of Christ is oral or sacramental, when the true, essential body and blood of Christ are also orally received and partaken of in the Holy Supper, by all who eat and drink the consecrated bread and wine in the Supper-by the believing as a certain pledge and assurance that their sins are surely forgiven them, and Christ dwells and is efficacious in them, but by the unbelieving for their judgment and condemnation, [64] as the words of the institution by Christ expressly declare, when at the table and during the Supper He offers His disciples natural bread and natural wine, which He calls His true body and true blood, at the same time saying: Eat and drink. For in view of the circumstances this command evidently cannot be understood otherwise than of oral eating and drinking, however, not in a gross, carnal, Capernaitic, but in a supernatural, incomprehensible way; [65] to which afterwards the other command adds still another and spiritual eating, when the Lord Christ says further: This do in remembrance of Me, where He requires faith [which is the spiritual partaking of Christ's body).

Then also Sections 73 and 74

[73] Since a misunderstanding and dissension among some teachers of the Augsburg Confession also has occurred concerning consecration and the common rule, that nothing is a sacrament without the appointed use [or divinely instituted act], we have made a fraternal and unanimous declaration to one another also concerning this matter to the following purport, [74] namely, that not the word or work of any man produces the true presence of the body and blood of Christ in the Supper, whether it be the merit or recitation of the minister, or the eating and drinking or faith of the communicants; but all this should be ascribed alone to the power of Almighty God and the word, institution, and ordination of our Lord Jesus Christ.