To which extent are people’s perceptions of their own pronunciation influenced by the language’s orthography?

Yes, there have been many studies of this. A quick search of Google Scholar will reveal many references.

This article on non-word processing has a good overview of the literature:

There is now considerable evidence that orthographic information is activated during the processing of verbal utterances, whether this automatically occurs during speech recognition (e.g., Slowiaczek et al., 2003; Chéreau et al., 2007; Taft et al., 2008; Peereman et al., 2009) or is controlled strategically (e.g., Cutler et al., 2009).

So, your anecdotes have a good experimental foundation to them. However, I think there's a huge value to collecting anecdotes such as yours, as well. They give examples of how the interface between orthography, phonological representation and normative judgements which would be hard to reveal in a controlled environment. Some other anecdotes, I've found revealing:

• Czech speakers who refuse to believe that 'kde' is correctly pronounced as 'gde' ([g] in syllable initial position is perceived as alien to Czech) or 'jsem' as 'sem'.

• English teachers who unproblematically syllabify 'kitten' as /kit.ten/

• The move to 'of' as an alternative auxiliary in the present perfect in British English. People will now enunciate things like "I should OF known"

• Speakers saying things like 'They don't pronounce the G in going' even though no 'g' was pronounced in going or goin'

• Here's one I've observed in myself and a few other Czechs speaking English which I think is very interesting. Pronouncing 'much' as 'match'. This is a whole chain of influences 'much' is first represented in Czech orthography as 'mač' which as an 'a' which is then 'read out' as if in an English word resulting in 'match'. (Extremly uncommon but I've observed this twice in myself and at least 2 other times in others.)

All of the above would be really hard to study in any other way than a systematic collection and analysis of anecdotes but I'm not aware of a study that has done that.