My understanding is:
Haemoglobin is a protein molecule in red blood cells. Lower than normal number of red blood cells means you have anaemia and the haemoglobin level reflects this number. A low haemoglobin level is also referred to as anaemia or low red blood count. So in other words both tests say the same thing in respect of anaemia.
I guess they are done separately to rule out other things - for example I think (though am not sure) that if you have a normal level of red blood cells, but low amount of haemoglobin, you could have conditions such as thalassemia (not sure this is as simple as this/completely accurate, but just trying to explain why there are two tests and results). I'm sure there are lots of things ruled out by having both tests though.
I am also not 100% sure whether there is a clear diagnosis level that makes anaemia 'severe' - but I know of several friends who have been told they have significant anaemia and they have had to have injections rather than tablets/liquid - so I would presume that your son is in the more moderate group as he is just prescribed medicine- though he is anaemic. I would also say not to worry too much - it is quite common and often undiagnosed, so the fact your son is being monitored and receiving treatment is a good thing.
How long medicine will take to work is on a per person basis.
One tip though is to also ensure he gets vitamin C (eg drink of fresh orange juice when he takes his medicine?) as Vitamin C helps the iron to absorb. I was anaemic as a teenager (when I went vegetarian - still am 32 years later!) and my GP told me to make sure I had a bowl of fortified cereal for breakfast every morning and to drink a glass of fruit juice with every evening meal to absorb the iron from the food with the vitamin C from the drink. This is really the only change I've ever made to my diet and I have never been anaemic since, and throughout both my pregnancies, despite being repeatedly hospitalised with hyperemesis, I had higher than average iron levels throughout - so it seems to work and is easy to incorporate.
My daughter was a very very fussy eater and we had her checked for vitamin deficiencies after school raised concerns when she was about 6 about her limited food intake (variety not quantity). She was fine on all counts and I really think the breakfast cereal every morning (and we gave her dried cereal pieces - eg malties/mini Weetabix/cornflakes/etc - as snacks instead of sweets etc) and she has a glass of smoothie with every evening meal even if the food is limited, is what made the difference. Cereal is also a good 'quick' choice for a fussy eater when others are having foods like pizza etc that my daughter won't eat - and tbh, the bowl of cereal is probably better for her then many 'fast foods'.
Are there any breakfast cereals your son will eat (most are fortified with iron so it doesn't really matter which one - just check the label) and any fruit juices? By the way, my daughter would only eat some cereals 'dry' (no milk) so we just let her - it seemed to bother other people when they saw it, but I really (still) don't know why it's a problem.
Looking back I think a lot of my stress about her 'fussy' eating was because of pressures from other people making comments, so if you are getting stressed about your son's selective eating, although it can be annoying and inconvenient at times, do ignore everyone else. (Oh and people telling me to make pictures and faces etc from the food to make eating 'fun' used to drive me up the wall - sorry, but my daughter was intelligent enough to know that a carrot mouth was still a carrot and she didn't like them!!)
Hope his iron levels start to improve soon. Best wishes x