I don't see much point in using eth and thorn for voiced and voiceless fricatives, respectively, for pedagogical purposes (though people may do so), as using only one forces the learner to understand the allophonic system of voicing crucial to OE.
Based on my experience of learning Old English, I would say the opposite happens, unless you're learning with a teacher who can spot mistakes early, or you have a audio recordings of
someone with good pronunciation. When I read texts that haven't been adapted for the learner I'm pretty good distinguishing long and short vowels and diphthongs and how to pronounce c and g. I put this down to the fact that I learnt from Sweet's Anglo-Saxon primer which uses dotted cs and gs and macrons to show vowel length. In the midst of learning all the other weird and wonderful things - dative, subjunctive, vocabulary, word order - I missed, or quickly forgot, the section on how to pronounce the lingua-dental fricative and unconsciously developed my own pronunciation. I put this down to the fact that Sweet uses only the thorn for both sounds.
I've since re-read the rules for the lingua-dental fricatives, but it's proving difficult to shift old habits. I wish Sweet had used the icelandic system of using ð for voiced and þ for unvoiced because I believe that I wouldn't have this idiosyncratic (probably) unhistorical pronunciation.
I had no trouble going from the standardized artificial teaching orthography of Sweet to other texts that are more faithful to the manuscripts: it's easy to recognize a word that has a different spelling than one is used to.
Other Languages' influence on EnglishI've noted a few times that Modern English word order is much closer to the Scandinavian languages, including old Norse, than it is to Old English or it's closer relatives. It also forms phrasal verbs on the Norse pattern where the particle is separated from the verb. Old English didn't do this and neither do any of the West Germanic languages. The tense system is practically identical to Continental Scandinavian languages and very similar to Norse.
I read somewhere that roughly 80% of English vocabulary is not from Old English, but that 80% of the words we use on a day to day basis is that 20% of Old English derived words. However, it seems that Norse has infiltrated the ordinals with 'first' which seems like one of the parts of the vocabulary that would be most resistant to outside influence (Romance got in there too with 'second'), and Norse even managed to make it into the pronoun system which I guess would be the most resistant part of vocabulary to outside influence.
For these reasons, I've wondered if English should be referred to as Anglo-Norse. Something like Norsified English with an elaborate head-dress of words from languages from all over the world including seven words of inuit origin (I can remember only 'anorak').
Spreading Old English Knowledge / Fighting FallaciesI've never had to argue about Old English. Every time I've talked with someone about pre-conquest English history, it's been met with a kind of "Oh, I never knew that, how interesting" type of response, but then I've never got into a discussion with an 'expert'. My guess is the 'experts' don't want to lose face when confronted by a 'non-expert'.
I would echo what (I think) Bowerthane said - "know your stuff' - something that I'm still working on, but I would imagine that 'experts' challenged on something they have got wrong are always going to act with disdain and contempt towards the 'non-expert', and we're just going to have to take a deep breath, present the evidence and walk away. Hopefully the 'experts' will check facts later, and quietly change their opinions and pretend that they always had those opinions. If not, we work on everyone else and try to build up the numbers till the telly people respond by making documentaries about the pre-conquest era and mini-series set in the pre-conquest era.