Disclaimer: This answer is from a Charismatic Evangelical perspective with Wesleyan soteriology.
Is receiving the Holy Spirit the completion of Salvation?
The short answer is no.
While it is true that Ephesians 4:30 ESV says
And do not grieve the Holy Spirit of God, by whom you were sealed for the day of redemption.
Given this, it would be easy to think that if we've received the Holy Spirit, we will definitely be kept from falling away and will surely be saved from the wrath to come (i.e. 'once saved always saved'); However, just a little further along in the same discourse, the believer (who has 'shared in the Holy Spirit, and ha[s] tasted the goodness of the word of God and the powers of the age to come' - Heb 6:4-6) is solemnly warned:
5 For you may be sure of this, that everyone who is sexually immoral or impure, or who is covetous (that is, an idolater), has no inheritance in the kingdom of Christ and God. 6 Let no one deceive you with empty words, for because of these things the wrath of God comes upon the sons of disobedience. 7 Therefore do not become partners with them; - Ephesians 5:5-7 ESV
This is not an idle warning, there are many places in scripture that believers are warned of the dangers of falling away (cf. Heb 3:12-14, Rom 8:12-13, Rom 11:17-22, Matt 7:13-27, John 15:1-6) - even given the unfailing faithfulness of the Heavenly Father, even after evidence of His mighty anointing upon us with signs and wonders. The simple fact is the Holy Spirit's influence in our lives is resistable (cf. Acts 7:51, Eph 4:30, 1 Thes 5:19) and the consequence of this, is that like Esau it is possible to despise our (new) birthright. As we receive Him, so we must walk in Him (Col 2:6).
So when we initially receive the Holy Spirit it is not the 'completion' of our salvation. Paul's instruction to "work out your own salvation with fear and trembling" (Phil 2:12) was given to believers who had received the Spirit. The Holy Spirit will lead us into the fulness of salvation, but we must follow and not shrink back through unbelief. The journey of sanctification is along the straight and narrow road that is difficult, but at the end of the road, our journey will be completed in full satisfaction of that which the Spirit's presence gives us a foretaste.
The 'completion' of our salvation is described here:
28 And now, little children, abide in him, so that when he appears we may have confidence and not shrink from him in shame at his coming. 29 If you know that he is righteous, you may be sure that everyone who practices righteousness has been born of him.
3:1 See what kind of love the Father has given to us, that we should be called children of God; and so we are. The reason why the world does not know us is that it did not know him. 2 Beloved, we are God's children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is. 3 And everyone who thus hopes in him purifies himself as he is pure. - 1 John 2:28-3:3 ESV (emphasis added)
The completion of our salvation, is to be conformed to the image of the Son (cf. Rom 8:29, 2 Cor 3:18, Eph 4:13-15) and to be perfect as our Father in Heaven is perfect (Mat 5:48). "Character trumps Charisma" (cf. Mat 7:15-23).
Nevertheless, receiving the full measure of the Spirit promised to the Children of God is extremely precious and should not be despised - He will convict of sin, of righteousness and judgement (John 16:8), He will lead us to confess that Jesus is Lord (cf. 1 Cor 12:3), He will guide us into all truth (John 16:13), He will equip us to be powerful witnesses to the Truth* (Acts 1:8), He will bring prophecy, dreams and visions that we may know the day of the Lord's favor and see His deliverance* (Joel 2:28-32).
The question correctly cites verses from Luke 11 and Acts 1 & 2 as giving keys as to how to receive the gift of the Holy Spirit - Ask, Seek, Knock, Expect the good gift from your Heavenly Father, Wait on God according to his timing, unite with other believers in prayer with one purpose. We see from the many other examples in the book of Acts that those who have received can help others to receive, particularly through the laying on of hands (cf. Acts 8:14-17, Acts 9:17-18). We can see further, that *'the baptism of the Holy Spirit' is a subsequent experience to believing the gospel and it was always evidenced by outward signs (gifts of prophecy, tongues or healing etc. - cf. Acts 8:14-17, Acts 9:17-18, Acts 19:1-7)
In closing, to answer your final question:
The question that arises in my mind is whether this receiving of the Holy Spirit was only applicable, those early converts or is that applicable to all converts?
This is directly answered in the clearest terms possible by what the Apostle Peter says in Acts 2:38-39 -
38 And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. 39 For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.” (emphasis added)
The promise of the Holy Spirit is to everyone whom the Lord God calls to himself - even those who are 'far off'.