You might be best asking the unis directly about employment rates, though of course there's a risk of bias.

I did an MSc at LSHTM. Great place to study, and its courses are well regarded generally. I don't know about the Med Stats masters specifically.

Always aim for (i) peer review and preferably (ii) indexed. Don't worry so much about citations - that can always develop over time. I don't think anyone would judge an early career researcher adversely if their research had yet to be cited, but they might query the quality of work if it wasn't peer reviewed.

At this stage, if you come out of your PhD with a couple of first/second author peer reviewed publications, you're doing well. Conference proceedings (even if indexed) are generally less important, though that can vary by field. Which field/discipline are you in?

As for your second scenario comparison, PhD vs publications and MSc, it depends entirely what job you'd be going for.

The honours situation is a bit of an odd one. I've never heard of that happening before. Would it be possible to obtain some kind of letter (on uni headed paper) from a tutor or lecturer there to explain the situation? It might have more influence than your explanation alone.

Lab based PhD supervisors will definitely want you to have lab experience. Unfortunately your email back to Liverpool won't change that outcome, but it's worth remembering in future applications to really highlight every bit of practical experience you have. If you didn't do this for the one with the 15th May closing date, it might be worth asking if you can edit your application accordingly.

If this current tranche of applications doesn't yield anything, you might want to consider applying for research technician or research assistant jobs, to get additional practical experience on your CV. It can also be a way of getting a foot in the door at a particular uni for future PhD applications.

I would also consider getting other people to read your applications to make sure there is a clear 'story' of why you are applying for each particular PhD. It always helps to get other people's perspectives. If I'm honest, I found your description above a bit confusing! You need to sell your expertise to supervisors so they don't query the route you've taken from chemistry to neuroscience.

It would help you to have a clear audience in mind. A computer programmer wanting to get into analysis will have different expectations and requirements compared to a research scientist with domain knowledge who needs to develop his/her skills to be able to deal with increasingly large datasets.

That's a lot of topics. Are you proposing a short course? A course unit? A Master's degree? Too much breadth and not enough depth might not be helpful.

I agree with bewildered. Messing conference organisers around because you don't think their conference is 'good enough' for your work any more isn't very polite and could cause you longer term damage. If you've extended the work since the original submission, you could always try to submit the extended work to another conference, though only if you have the intention of attending. Don't forget though that conference proceedings are 'worth' less than full journal articles. If you really want your work out there and to have an impact, focus on writing full papers.

If I'm not crossing a line, it seems to me that you've opened a lot of threads of late querying a number of aspects of the supervisor-student relationship and expectations, and what to do about publications etc. Although you will of course find help here, I'd strongly suggest putting some effort into trying to develop a support network in your own department or field who can direct you to answers that might be more specific to your particular PhD journey. It sounds like your relationship with your supervisor is a little bit tense on matters such as publication - it really is worth trying to make this relationship work as well as possible. You don't have to like him/her, but you do need to find a way of utilising this relationship to your best advantage and least stress.