The post doc shouldn't put their name on work that was done before they entered the lab group, but any work done afterwards it may be seen as fair game to do so. This will be something you need to clarify with your PI.

What will the post doc be presenting at the conference? Your work or theirs?

What is the situation re: the post doc's access to your work/data? Is it only the abstract they have used so far?

It might be a good idea to sit down with your current PI to discuss what the post doc's exact role will be, what they will have access to, what they will get priority over etc. Politics around author order will always be a sticky situation in academia; this post doc obviously thought it is entirely acceptable to place their name above yours, whereas you don't think that is fair. This is something you may need to discuss together. Depending on the situation/relationship you have with your group, sometimes authorship order is by contribution, by surname alphabetically, or by superiority.

I've recently spotted a relevant research opportunity in my discipline, and it's all online due to COVID-19. That's great for me as I don't live near the particular university offering the role, and I am self-funded.

However, I am slightly concerned about the number of hours the role will take per week (as I am doing my PhD full-time). I would have to work every day for 3.5 hours, as well as some Saturdays. I'm worried that my own project might suffer, although the position is only for 3 months and therefore wouldn't be forever. I also don't know how happy my supervisor would be about it.

Has anyone been a research assistant during their PhD? How did you find it in terms of workload etc.?

It depends on the area of Linguistics you are interested in - if it's an area which a large Psycholinguistics component, your current background may be enough. Tbh, Humanities PhDs are competitive funding-wise, so a lot of research councils ask for specific previous degrees - you might want to research potential supervisors and funding sources to see whether the route is viable for you.

There is also the MA+PhD route (1+3), which would fund a relevant masters degree plus the PhD, which might be suitable for you if you have a less than ideal academic background for Linguistics.

I agree with Tudor_Queen. Research assistants who are fluent in the languages you're analysing would be best, as translation software is never that reliable or accurate.

It depends on the field, but some research councils will not fund current PhD students, only students will are yet to start their project.
Also, I'm not sure what you mean by looking for a fully-funded project for next year. If you've started a PhD elsewhere, you may find that it's frowned upon to up and leave after 1 year, and is a bit of a waste of time and money honestly.

Your mindset seems pretty persistent, so I think seeking mental health support is very important at this stage. If you need to apply for an extension in order to complete your project satisfactorily, then look into doing that. If you've had to restart your project in your 3rd year, that definitely needs addressing - by your committee, and by the university.

I think a big question for you is: what would make you feel less incompetent? And how could you go about achieving that?

I can relate, not at PhD level, but some things to do with my undergraduate degree can give me a very visceral response. Like you, I received very little support during my undergrad (even when I actively sought it), and felt like I wasn't wanted. My particular course had a small cohort, so it was easy for people to stand out for the right or wrong reasons. Only a few weeks ago I was reading an academic paper about a certain topic highly relevant to my undergrad, and I found that I suddenly felt very tearful and panicky. I graduated in 2014, so I was surprised to still feel like that.

What helped me was making sure that every career decision I made from then on came from a place of seeking happiness overall, and not doing anything as a means to an end. It also gave me a good measure of who are good people for me and who aren't, and to trust my gut instinct more.