I would like to know whether it is possible to be saved in the Catholic Church, and whether or not members of that church are truly following Jesus Christ.
The best way to approach this question might be from the other side first. As I know for a fact (from discussions with Catholic friends and supporters), the Roman Catholic Church would almost certainly object to the vast majority of teaching at Ichthys.com. Given this fact, and given the R.C. doctrine of extra ecclesiam nulla salus (i.e., "there is no salvation outside of the Roman Catholic Church"), we should start by stipulating that many of my Catholic friends and acquaintances (at least those who are fairly strict adherents of Church dogma) would be likely to respond in the negative were a similar question to be asked about myself and those who follow this ministry.
So I suppose that it stands to reason that the teachings which I believe to be true (as contained on the website) contradict Catholicism in very large part - if I believed that the R.C. Church was right in their opinions, I would be R.C. (and, in a similar way, Luther, Calvin and the other reformers would never have left that church).
Catholic dogma is incredibly extensive, and I certainly would not wish to say that I have acquainted myself with every facet of it - far from it. So let us stick to some of the main principles. Some of the major objections that early Protestants had with the dogma of the Roman church as it developed over the centuries, I and many non-Catholics (especially evangelicals) would still see today as major errors, for example:
1) salvation by works (i.e., the idea that "doing good" can atone for sin or appease God). But in truth salvation comes through faith in Christ alone by the grace of God, and any work done apart from the power and will of God is of no account (Eph.2:8-9).
2) the vicar of Christ (i.e., the idea that the Pope, R.C. officials, priests etc. are God's substitutes and have God's authority). But in truth every believer has equal access to God through Jesus Christ and has a direct relationship with Him through our Savior independent of any other human being (Eph.2:18; 3:12).
3) the authority of the institutional church (i.e., the idea that it is not the Bible which is the sole source of truth, but what the R.C. church says is the truth). But in truth, Jesus is the truth and His Word is the only truth we can know, irrespective of the traditions of any group (Jn.14:6).
It is on this last point that I believe it is easiest to split true believers and those who are merely devoted adherents to Catholic doctrine. I always tell everyone, Catholics included, that if one is really trying to seek God, then keep reading the Bible, and God will honor your genuine desire to know Him, and will use His Word of truth to do so. The R.C. church (and doctrinaire adherents to it) are very clear that they do not feel the Bible to be the final authority, and that they do feel that it is important NOT to pay attention to the Bible where it disagrees with R.C. doctrine (which, in my view, it does on nearly every page).
So to make a long story short, in my simplified view, the essence of Roman Catholic dogma is faith in an organization (for salvation and everything else), whereas true Christianity is faith in God and Jesus Christ the living Word, through His teaching, the written Word (independent of any human organization).
Having said this, I would wish to make two further important points, without which the context of the above remarks may be easily mis-construed:
1) as in many other groups, just because the group officially thinks a certain way, doesn't mean that there aren't some in the group who have their own way of thinking. At seminary, one of my professors, a former Catholic, was adamant about the fact (as he saw it) that salvation within that church was impossible. Personally, I am unwilling to sell God short on this: there may very well be individual Catholics who are true believers in Jesus Christ and counting upon their faith in Him for salvation (rather than works, rather than their membership in the R.C. church) - only God knows the heart. But it is certainly true that having anti-faith doctrines at the core of their dogma makes it necessary for such individuals to ignore much of the church's teaching.
2) on the other hand, since it is not only in the Catholic Church where one finds individuals who are contrary in their thinking to the official beliefs of the group, we are sure to find a number (possibly a large number) of members of non-catholic groups, even doctrinally sound evangelical groups, who are in no way true believers (even though the essential teachings of their group about salvation may be correct).
For it is as Paul said about his fellow Jews vs. gentiles in Romans chapters 2-4: it is not race or birth that brings salvation, but response to the truth in one's heart (for we are saved by grace through faith: Eph.2:8-9). Therefore, since it is not any organizational affiliation which determines our heart, our relationship with God is based upon how we respond to Him, His Son, His truth in our heart, and not by externals such as church membership (whatever the church), or the "good" works we may do (even assuming they are truly good and properly motivated). It is important what we belong to, whom we listen to, to what we give our allegiance. One of the most difficult tasks confronting believers of our day who truly desire to follow God with all of their heart is exactly where to belong - so much of contemporary Christianity has been so badly compromised. And if the R.C. Church has codified principles which are demonstrably and unashamedly anti-biblical (i.e., the teachings and pronouncements of the organization trumping the Bible), it is also true that there are many, many other so-called Christian groups which are in reality are no longer really very concerned with what the Bible has to say either (except in the most superficial of ways).
Well, apologies for failing to do this question the justice it deserves. Let me end with the final part of your question about a Catholic calling him/herself a "follower of Christ". To this I would say, many people in the history of the Church from all sorts of groups have called themselves this and have not truly been so - the question perhaps is not so much whether one can say such a thing, but rather whether or not it is true. For God knows all who are truly His (2Tim.2:19). Whoever seeks Him in truth, will be lead to the truth (Lk.11:9; 12:31); that is true whatever organization or situation a person finds him or herself in. And for all those who truly are seeking God, God will not let them down, but will lead them, prune them (Jn.14), and draw them out of any and all pernicious circumstances that are stunting their spiritual growth - if only they will persist in seeking and following Him.
God makes the issue clear to us all - sooner or later we shall all come face to face with the truth, one way or the other. For in truth there is really only one Way, faith in and faithful following of the Master who bought us (Jn.14:6).
You may also find the following links helpful:
Red Hot or Lukewarm? Bible Teaching versus Sermonizing.
Aspects of the False Doctrine of Institutional Security.
Three False Doctrines that Threaten Faith.
Mega-Churches, Emergent Christianity, Spirituality and Materialism.
Characteristics of the New Religion of Antichrist.
In Him who is the only truth, our Lord and Savior Jesus Christ.