Is there a terminology contradiction about whether the conjugate of a strong acid is a "weak base"?
(Would rather have made this a comment than an answer but still new. So please don't hang me.)

Through high-school, I was also bugged by this. No, you are not going mad, there are many different views on this matter when you google it. The problem is that like many other scales humans create, the way we define 'strong' or 'weak' is a rather relative concept, especially when it gets to the mid-ranges. It is only a way for us to explain what we observe and get a handle on things. 

One thing all chemists would agree on is that you should rather look at how strong an acid or base is relative to another (based on Ka or Kb values). In the end, these are the things you would actually use. However, as you need an answer in a school context I would suggest the following:

At school (at least where I come from) we mainly dealt with Bronsted-Lowry acids and bases. Here the strength of an acid depends on how easily it donates ("accepts" for bases) protons. If your students already know a bit about Ka values then anything with a Ka>1 can be considered (reasonably) strong: on a school level, this is usually limited to HI, HBr, HCl, HNO3, H2SO4 and sometimes HClO3 and HClO4. You might want to look at the relative strengths of these 'strong acids' (I seem to recall an exam question about why HI is a stronger acid than HCl (how stable are the ions) and why HF isn't considered a strong acid - the answer is H-bonding). Anything with Ka&lt;1 can be seen as weak.

As to conjugates. Your statement in 2 about 'The stronger the Acid the weaker the conjugate base' and its inverse applies. Here you also work comparatively rather than absolute. You could try explaining it by saying if something gives away a proton easily (acid strength) why would it want to take that proton back later on? Mathematically this can be expressed by relating the Ka value to the Kb value of its conjugate Ka = Kw/Kb (Kw = 1e-14). You would notice that very few 'weak bases' (Kb&lt;1) will give you a Ka>1, from there the confusion about "the conjugate of a weak base is a weak acid". eg. H2PO4^- (Ka = 6.23e-8) is neither a particularly strong base nor acid, but if it reacts as a base and forms H3PO4 this conjugate still only has a Ka = 7.52e-3 - which is still very weak compared to HI.