Chronos - I understand your objections to the argument, but feel they can be overcome. Mostly they seem to be related to problems of language.
1. Something is contingent.
This assumes that something that is contingent exists. This may be true or untrue depending on what exactly is meant by 'exist'. Kitchen tables can be said to exist, but they are epiphenomenal on the particles and waves that constitute them, and the spacetime that contains them, and so on. So there is a sense in which we can say that waves and particles, time and space, exist in a stronger sense than do kitchen tables.
If we give the word 'exist' its very strongest possible ontological meaning then all epiphenomena do not exist, and all that truly exists is what is not-epiphenomenal (not-dependent, not-contingent). This is the way 'exists' is used in Buddhist texts, (with one important proviso that doesn't matter here).
In this view all the phenomena that appear to us to exist are in fact epiphenomena (contingent/dependent), and there is only one phenomenon that truly exists, that is not contingent on something else. So in this view, at the deepest level of analysis, the first statement in the argument premise is not true. However things that are contingent/dependent do appear to us to exist, and in this sense the statement would be true.
2. If something is contingent, its ultimate cause is either self-caused, uncaused, itself merely contingent, or a necessary being.
3. Its ultimate cause is either self-caused, uncaused, itself merely contingent, or a necessary being.
I have no problem with these two, except would prefer 'substance or entity' instead of 'being', since to say 'being' is to make an assumption.
4. Its ultimate cause is not self-caused.
5. Its ultimate cause is not uncaused.
6. Its ultimate cause is not merely contingent.
The example of self-causation backwards and forwards through time seems incorrect to me. It's reminiscent of the Wheeler-Feynman model of time in which advanced and retarded waves cause events in the past and the future. The problem is that even in these kinds of models something has to start the causal ball rolling, and self-causation seems to be equivalent to non-causation. So I'd go along with statement 4.
All these three statements suffer from their wording which, on the surface, seems to just assume that there is such a thing as "Its ultimate cause". However I feel this is just a problem of language. It could be re-written "The ultimate substance or entity from which which the universe arises is neither self-caused, uncaused or contingent". That's how I read it anyway.
7. Its ultimate cause is a necessary being.
Perhaps the argument shows that all phenomena except one must be contingent and that this ultimate non-contingent phenomena must be necessary. However it seems like jumping to conclusions to call it a being.
The question I have is regarding premise 6. What does it mean for something to be "merely contingent"? In the first premise, "something is contingent" it is meant that there is something here that did not have to be here.
I take 'contingent' as equivalent to 'dependent'. That is, a thing which has "merely" a contingent existence is something that could not exist withour dependence on other phenomena. Thus a kitchen table has merely a contingent existence, as does gravity, mass, motion, baseball, colour, human beings and, according to some, everything else that presents an appearance to our senses or intellect. (Kant seems relevant here, with his 'transcendent reality'). Something which is not contingent can 'exist' while being the only thing that does.
The argument as given shows, or sets out to show, that something must have a non-contingent existence and that this is God. The first conclusion seems reasonable, since it would be odd if there were more than one fundamental non-contingent thing, and odd if there weren't one at all. However to show that this thing is God would require a new and different argument, and a very clear definition of 'God'.
So in premise 6, is this "merely contingent" ultimate cause that is being ruled out to be thought of as a thing that has existed forever, or are we talking about a member of an infinitely backward-regressing series?
This is the paradox. Neither answer makes sense. It is a metaphysical question and as such is undecidable. There is no way out of it, except perhaps Lao-Tsu's nondual "causeless cause" or "Tao". The Tao is not a cause (and is not not-a-cause!) but could perhaps be said to supply the contingent condition necessary to the existence of epiphenomena, and thus of the universe. (To a Taoist scientists study only epiphenomena, things which are not ultimately real).
Intellectually this fundamental thing must be conceived as being either a cause or not a cause, for what else could it be? This is a case of "tertium non datur". It is inevitable that according to reason it must be one or the other, a cause or not a cause. This is why we have metaphysical questions (and discussions like this). Metaphysical questions embody the assumption that one of their answers is right and the other wrong.
But a Taoist would argue that this is dualism, and the cause/not-caused dilemma is simply evidence that this ultimate non-contingent 'thing', God or whatever, cannot be conceived but can only be known directly, and also evidence of the limits to reasoning, the impossibility of encompassing within our two-value systems of reasoning something which is ontologically ultimate but which is neither a cause nor not a cause. To do so would give rise to all sorts of inconsistencies in any system of philosophical reasoning.
A Taoist would say, I think, that because of all this if this ultimate thing is to be represented in any system of reasoning it must be by way of an undefined term, a term that points to the thing but which implies nothing about it beyond its undefinability. This term could be 'Tao' but there are many others. I think I read somewhere that Sufis have 99 official names for it.
This thing/not-thing is presumably something like a 'wavicle', neither a particle nor a wave and thus impossible to conceptualise in terms of particles and waves. "Incomprehensible to us" as Richard Feynman says of them. If you look you'll find that the term "Tao" plays precisely the same role in Taoist epistemology as the term "wavicle" does in the epistemology of QM.
Because of the existence of this other view (Taoism etc.) the argument you were given by your tutor would, in my opinion, represent an argument not just for God, but also for the Tao, Buddha-nature, Allah (in Sufism anyway), Unicity, and many other terms used to denote the ultimate non-contingent phenomenon. Therefore to me it does not represent an argument for the existence of God.
However, whether this thing can be called God depends only on the definition of God, and it can get confusing. While probably agreeing with the argument you gave in outline at least, as a general rule a Sufi would argue vehemently that Allah is not God, a Taoist would argue that the Tao is not God, Buddhists would argue that Buddha-nature is not God and so on.
Also, it's a slightly strange argument. It sets out to show that there is something that exists necessarily, something that is not contingent. It then simply assumes that having done this the existence of everything else is explained. In other words it assumes not just that God (or whatever) is non-contingent and exists necessarily, but also that God is not caused but is causal. This is counting chickens before they've hatched. It may be logically inevitable that what is ultimate is non-contingent and necessary, but it is not at all obviously reasonable to suppose that something that is uncaused can be causal.
This issue has had some attention recently because this is how scientists think of consciousness, that it is caused but not causal. The same basic idea with the opposite polarity. (It may be just coincidence of course, but it's odd that it is God (or whatever) and consciousness that turn out to be the only two things in the scientific model of the universe that have this lopsided property). This idea, which is known as "assymetric supervenience" in consciousness studies, makes little sense to me. I can't make a knock-down argument against it because the issues are too complicated, but there have been one or two good ones in the literature.
I hope that made some sense.
Absolutely, it's an interesting question. I hope this makes some sort of sense also.