Hi There!
Learning a sax is not an exact science - it can be trial and error - so what I have said is subjective. In my experience (I have had a total of 9 teachers in various instruments) I would say that the majority of those teachers knew very little about equipment, and only 2 offered any advice about equipment - which mouthpiece sounded better in both cases. However, they did know all there was to know about playing said instrument etc.
There is a commonly accepted wisdom that you should stick with a set mouthpiece for a year or more and not "chop and change" which I think is potentially flawed. I would argue that it is better to have some alternatives in order that you can make a more informed choice of what you play/learn on. Not every mouthpiece is right for everyone, not every reed is playable for everyone, and having an alternative choice can help enormously. Hence having a cheap alternative mouthpiece to your Kessler might be really helpful, just as having an alternative strength reed might help clarify a few things also, at minimal cost.
Regarding Facing/Tip opening - this refers to the gap between the tip of the mouthpiece and the tip of the reed. The larger the gap the harder to play but louder and possibly larger range of expression. The smaller the gap the easier to play (unless too small) and quieter, usually more precise sound produced. People who play larger tip openings often use softer reeds. If you look on the Jody Jazz Comparison Chart - on the Jody Jazz website - you will see the sizes of the various manufactures facings/tip openings. Many manufacturers use slightly different numbering systems - hence a Runyon #4 is smaller than a Rico#3 - thus the chart can help by spelling out the actual size in thousands of an inch (or centimetres in the case of some European brands like Vandoren/Francois Louis).
So I would recommend trying out various combinations of mouthpiece/reed to see what feels most comfortable at this stage, and to narrow down what may be affecting your note production at this stage. Practising long tones is excellent, but you do need equipment that feels comfortable at this stage, so I stick by my recommendation, and make no assumptions whatsoever that all teachers are clued up on equipment etc. I play trumpet, and started on the Bach 7C mouthpiece (16.20mm) recommended by virtually every trumpet teacher. I used to be frustrated that my range of notes was not what I liked it to be. I bought a Bach 1 1/2C (17.00mm) on ebay - it was cheap - and suddenly found a much wider, more expressive range of sound than with the 7C. I currently play between 17.25mm & 17.75mm mouthpieces, but struggle with anything smaller.
A Rico mpc will cost you less than $17, and a Runyon 22 about $30, and you can get reeds in packs of 3. I am not recommending endless chopping and changing of equipment but having an additional mouthpiece (smaller) and reed (softer) will give you 4 combinations & not just one. Imagine that you had to make do with size 8 shoes when you were actually a size 7.5 or even 7.
Kind regards
Tom