Do the reformed theology adherents believe that a believer should not "converse with" or "pray to" or worship the Holy Spirit?

I do not know who it was that organized your house group, but the churches that call themselves "Reformed" hold to the same doctrine of the Trinity that virtually all other Christians do. In this doctrine the Holy Spirit is one person of the Trinity, fully God and fully equal with the other persons, making it perfectly OK to pray to and worship him.

There are several branches of Christian Reformed church, and here is a statement of faith from the Reformed Church of North America, as an example. Here is another from the United Reformed Churches of North America.

Reformed theology maintains the full deity and personality of the Holy Spirit, so that there is no inferiority attached to the Holy Spirit that would result in the kind of subordination that the ‘Reformed House Church’ you mention makes in its statement of faith. What they claim seems to be but one step removed from ancient movements such as the pneumatomachians, or Macedonians, who denied the deity and personhood of the Holy Spirit. They were refuted by the Cappadocian ‘Father’ Basil who I will quote later.

At the outset, I will quote from Reformed theologian Michael Horton in his 2011 seminal work, “Pilgrim Theology” (Zondervan, abridged from ‘The Christian Faith’). It is important to understand why, in this century, some Christian groups have moved so far away from the reverence for the blessed Holy Spirit that was held by the early Church, that we get statements such as the one you have stated in your question. Once we grasp what the current theological state is of apparently ‘Reformed’ groups who verbally claim the Holy Spirit to be divine and personal, yet say he is not worthy of worship, then we will see why they are but one step removed from ancient groups denounced by the ‘early Church Fathers’.

In his chapter on the Holy Trinity, subheading “III. Practical Benefits of the Doctrine of the Trinity” Michael Horton writes:

“The Father, the Son, and the Spirit stride across the chapters of redemptive history toward the goal whose origin lies in an eternal pact between them. We worship, pray, confess, and sing our laments and praises to the Father, in the Son, by the Spirit. We are baptized and blessed in the name [singular] of the Father, the Son, and the Holy Spirit… We are adopted as children, not of a unipersonal God, but of the Father, as coheirs with his Son as Mediator, united to the Son and his ecclesial body by the Spirit… As we noted earlier, “to the Father is attributed the beginning of activity, and the fountain and wellspring of all things; to the Son, wisdom, counsel, and the ordered disposition of all things; but to the Spirit is assigned the power and efficacy of that activity.” [Calvin ‘Institutes’ 1.13.18.] No less than the Father are the Son and the Spirit our Creator and preserver. No less than the Son are the Father and the Spirit our Savior and Lord. No less than the Father and the Son is the Spirit ‘worshipped and glorified.’ “One of the reasons that many Christians have found little practical relevance of this doctrine for their lives is that our public worship – and therefore private piety – has become increasingly emptied of Trinitarian references. As we’ve seen, one of the reasons for the controversies and greater refinements in formulating this doctrine is that monotheistic Jews were now offering worship to Christ and the Holy Spirit as well as to the Father. In addition to the New Testament formulas for baptism and benedictions, ancient prayers and hymns planted the Trinitarian faith deep in the hearts of Christian people across many times and places. Christians throughout the ages didn’t just talk about the Trinity (which still, more often than not, happens today), but to the Father, in the Son, by the Spirit. “Many forms of worship today, however, have dispensed with these rich resources without replacing them with equally Trinitarian elements… To the extent that our experience is not Trinitarian, it is not properly Christian. One of my goals in t his book is to explore the relevance of the Trinity not only across the whole system of Christian doctrine, but in our lives as worshipers and disciples of Jesus Christ. “Many of the differences in faith and practice between Christian denominations and traditions can be attributed at least in part to a tendency to overlook this mutuality of the three persons in every work. It is not surprising that liberalism reduced the Trinity to the Father (as in Adolf von Harnack’s oft-repeated formula, ‘the universal fatherhood of God and universal brotherhood of man’) and therefore has had little interest in redemption by a divine Savior or its supernatural application by the Spirit. Deism needed only an Architect, not a Contractor and Builder. The tendency to focus on Christ apart from the Father and the Spirit has also led to a reductionistic view of redemption that is disconnected from creation and consummation. Placing the Spirit at the center – often in reaction against these other tendencies – one can easily treat the Spirit as a freelance operator rather than the one whose mission is to shine the spotlight on the Father’s word concerning his Son’s work. Throughout this volume we will be fleshing out what it means to say that in every external work of the Trinity all things are done by the Father, in the Son, through the Spirit.” (pages 103 to 104)

This lengthy quote shows why apparently ‘Reformed’ groups who verbally claim the Holy Spirit to be divine and personal, yet say he is not worthy of worship are but one step removed from ancient groups denounced by the ‘early Church Fathers’.

Here now is a little bit about the Cappadocian Father Basil the Great, in Roger E. Olson’s book, “The Story of Christian Theology” (InterVarsity Press 1999)

“Finally, Basil turned to Christian experience of salvation and argued against the subordinationists of the Spirit that since the Holy Spirit effects our salvation, he cannot be anything but God. Only God can save… In all things, then, the Holy Spirit is incapable of being parted from the Father and the Son. It is the Spirit who applies God’s salvation to our lives… Of course, Basil was more than willing to allow a certain kind of subordination of the Spirit to the Father as the Father is the eternal fount of all divinity from whom the Son is generated and the Spirit proceeds. The analogy is to the sun and its light and warmth. The latter originate in and from the former without being inferior or ‘after’ it. So the Son of God and the Spirit of God are God’s eternal counterparts sharing in his very being and glory while being subordinate in position but not in being to God the Father… In ‘On The Holy Spirit’ [Basil] declared theological war on those who would in any way deny the Spirit: “But we will not slacken in our defence of the truth. We will not cowardly abandon the cause. The Lord has delivered to us as a necessary and saving doctrine that the Holy Spirit is to be ranked with the Father” (Basil ‘De Spiritu Sancto’, 10.25) “Why (Olson continues) “was the deity of the Holy Spirit so important to Basil? … Basil would cry that to deny the Spirit’s deity is to place a question mark beside the deity of the Father and the Son. In Scripture as in worship as in personal Christian experience, the Holy Spirit is always associated with them as sharing equal honor and dignity, and equal honor and dignity imply equal nature. One cannot be ontologically subordinated to the others without that impinging on the honor and dignity and glory of all the persons of the Godhead.”

To say that the Holy Spirit is God, yet to decline to worship the Holy Spirit, is to make a mockery of the full deity of the Holy Spirit. It amounts to mere verbal declarations of honour, that are without any substance. I would say that Reformed literature is replete with evidence that the Holy Spirit is to be worshipped, prayed to, and addressed with all due reverence as to the Father and as to the Son. I have given but two recent examples of such Reformed literature (one of which harks back to the view of the Cappadocian Father, Basil.) I hope this indicates that the Reformed house church you allude to is on very shaky theological ground indeed, if it wishes to retain its ‘Reformed’ status.