let us clarify a few things:
heretics: Yes, heterodox Christians are indeed heretics - that is they hold a Christian belief that is at odds with the Orthodox faith (which we proclaim in our own prayers to be the "true faith" - see the hymn after communion in the liturgy). Heretic is not a "dirty word" or a "curse" or some kind of insult - it is a statement of fact - these Christians, however well meaning they may be, however nice they may be, however "inclusive" they may be are outside the Church. At the very least these protestants will generally deny the sacramental nature of the Church, the visible nature of the Church and the authority of Holy Orders (that is the authority of the priesthood to administer the sacraments).
praying with heretics: Praying with someone is vastly different than praying in the presence of someone or praying at the same time as someone. When we pray with someone, taking their words as our own or agreeing with their words as our own, we enter into their prayer, we pray with them. The ancient saying, "lex orandi, lex credendi" applies here. It means as we pray, so we believe. If you enter into the prayers of the heretics, you embrace their heretical beliefs - even if those beliefs are not explicitly stated in the prayer but implicitly a part of the form and structure of the prayer (and if you don't you become a hypocrite and liar by confessing as a belief that which you do not believe). Either way, it is not a good idea (and yes, sinful). You cannot remove yourself from every time and place where such prayer occurs and so it is best to learn the Orthodox prayers that have been given to us so that they become your own prayers, issuing forth from your heart without the need for written words. The Jesus prayer is a good start; the Trisagion prayers are also easily memorized. Look to the Psalter, the Divine Liturgy and others. Even the simply prayer "Lord have mercy" is appropriate in nearly any situation - repeat that prayer if nothing else (this is especially good in situations where people are expressing their concerns for others in their form of intercessory prayer. You can take the same concerns and simply offer the prayer, "Lord have mercy" committing the whole situation to God's providence.)
Participating in heterodox or "nondenominational" prayer meetings/Bible studies: While there is a significant "feel good" element here about being around other people who accept you for who you are (unlike God who accepts you where you are and then expects you to deny yourself, take up your cross and follow Him), one must be extremely careful in such participation. The "feel good" element is like the bait of a trap - it masks the danger lurking behind it and lures you further in until you are caught up in that danger. (I'm not saying your friends are the one's doing the trapping - the devil has set the trap and you are all victims together). If you tacitly approve of the trap by knowingly walking in and pretending that its all OK, then you facilitate their captivity (even if you manage to escape yourself).
You say that "During these meetings, we can study the Bible and sometimes we pray, if someone asks to pray (let's say if someone from them is sick or has a family member that is sick). Or we may sing songs to Glorify the God." When you study the Bible, do you all sit around and say what you think the Bible means or maybe share what a particular passage has meant to you? Do you accept the authority and teaching from someone who is teaching a different doctrine of that given by the Church? Not good in either case. To the first (what I think the Scripture means) I have to provide the response of a dear friend (a hieromonk well known to some of the old timers on this forum - Fr A) "The Church doesn't care what YOU think, we have the revelation of Christ and 2000 years of Holy apostolic Tradition to tell us what is true!" So if you are "studying the Bible" are you learning what the Father's teach about the Bible or are you wallowing around in your shared ignorance. Or perhaps you are using a book or study guide from some protestant teacher or preacher who is invariably teaching a "different Gospel from that which you have received from (the Church)" (2Cor 11:4) If this is the case, then you are allowing yourself to be led into heresy.
Prayer for others we already covered - on to "singing songs". What are spiritual songs, hymns, if not prayer. There is a lot of heresy disguised in the words of a song and it just kind of "slips by" out of our mouths without so much as a second thought. Only sing those songs which are given you by the Church. Remember that the whole of our services are sung ((they are a series of songs or even one long song) and the purpose of the things that we sing is to convey the true faith, to teach us the Orthodox faith. If its not an Orthodox song - then best not to sing it.
Finally you said "But in the recent years, I have read (and also my Spiritual Father has said to me ) that we must not pray with heretics, also I have read that this may be considered a self-excommunication from the Orthodox Church." So now you compound an innocent error (that of attending a prayer meeting) by willful disobedience to your Spiritual Father - thinking that you know what is better for your spiritual life than he. Disobedience is nothing more than self willed pride and such pride is the root of our separation from God. A side note here on "self acting excommunication" - there is no such thing. Excommunication must be imposed by the Church on a person - and so we cannot excommunicate ourselves apart from actively renouncing the Orthodox faith. However a person can put himself in a place where he is susceptible to excommunication, and it is this state which is often referred to (mistakenly imo) as "self-excommunication" The path is still open for one to repent and even when repentance is slow, the Church will hold the door open for one to correct his ways as long as possible.
So, stop being a self-willed disobedient spiritual child, intent upon walking into the devil's deadly trap because it feels good and the people are all so nice. Be an Orthodox Christian, which is to say, be obedient to the instruction and practice of the Church and especially to that of your spiritual father. It is not easy to work out your salvation, but the more your strive, the more God will pour out His grace upon you to help.
Fr David Moser