If you google posterior tendon dysfunction there is a long article about the difference,I have had tendonitis for ages but have always had a good level of function although foot was a bit flat.I have had loads of physio to build up other muscles and improve foot position...and although not 100% pain free the inflammation is now only in the foot insertion instead of all the way along the tendon .I am now back to running 5k since starting to build up very slowly in October. It has been a very long haul and I would advise someone to get to a physio who will look properly at bio mechanics early. I wasted a lot of time seeing physios who only treated the symptoms.
I used to have fairly regular sports massages and the therapist , who is also a qualified osteopath so knows her stuff, would make sure she did not go in too deep in any session a week before an event.
I have a fallen arch and wore orthotics successfully for a few years. Then forgot to wear them on a backpacking trip and damaged my posterior tibialis tendon as a result. Have since had heaps of physio focussing on 'listening foot' exercises which have enabled me to have a better foot position plus glute strengthening. So, I think one's foot position possibly can be improved and I am left wondering if I had taken this approach in the beginning rather than having orthotics maybe I would not have got injured? Who knows.
You do not need to be able to do a 5k in a certain time to do a 10k event. Just increase your long run once a week by no more than 10% until you are at 10k distance.
To speed up for a 5k, have you tried interval running? ie running a bit faster than you usually do for very short periods like 1 minute with a minute or two rest at a slower speed in between. Best to get more info on this and find something that works for you but this could easily be done on your treadmill runs.