A few comments and responses to your questions.
- going to Tribunal is costly, and you and your Solicitor will need to be confident you have sufficient tangible evidence to support your case, ie that you can prove they were aware of your MH condition during your appointment, and that even when you asked them for support they failed in their duty of care under the Equality Act.
Taking an employer to Tribunal is costly. There are no Tribunal fees (abolished in Feb 17) however legal fees can amount to £3,000+ esp if you go for legal representation at Tribunal. This is money you cannot get back even if you win. If you get an Award, it could be significantly eroded by fees. Your employer won't be paying out of their own pocket, you will. Do you have legal cover under your household insurance As you have an "event" to claim against ie dismissal, you could recover a large proportion of the fees if you win. Could your Solicitor do a no win no fee approach to mitigate cost?
1) Will this ruin my reputation in my future career? Will I be seen as a trouble maker if I try to sue an employer?
Not necessarily, if you have a good work history, focus on that with any new employer. Of course, you would never mention you'd gone to Tribunal - it's possible HR departments use the Gov.co.uk website, Tribunal Decisions, but you could write to the court and say you want to exercise your Right to be Forgotten under GDPR (from May 2018 onwards), so they take your record off Open access.
2) How long is this likely to drag on for? (The solicitor believes it will be settled out of court, but still...)
It could take 6-8 months. Several things must happen - you need to lodge your ET1 to start the process (your Solicitor should be guiding you on this) It
must be
before 3 months + 1 day since your dismissal. The court will then confirm the date of your hearing, which is often several months. Meanwhile your employer has to respond to your ET1 with their ET3. Also, at any point your employer could decide they want to end this, to mitigate their costs, and may make you an out of court offer, under a negotiated settlement. So it could be a lot quicker. But very often Respondents (ie employers) drag it out in the hope the Claimant (you) will back down.
3) Can my employer's solicitor's demand to see my medical records? I have nothing to hide, but I'm so frightened of them trying to bad mouth me / make me to sound like I was unstable etc.
They can request access to your Medical Records, your GP will never give them without your permission. You have every right to refuse, and you could commission a report from your GP costing about £75 which is a much better way of doing it because you are in control, and your GP will give strong evidence of your Medical condition. You could pay a visit to your GP and talk through what they could include in the report. Your Solicitor would submit this in your Court Bundle as a key aspect of your evidence.
Has your Solicitor suggested a merits assessment? This involves a barrister reviewing your case and assessing likelihood of success at Tribunal, ie is the case strong, do you have the right kind of evidence, how does your case compare to legal precedent (tribunal cases won and lost). This would cost c£750, but could save you the stress of taking a weak case to Tribunal and coming away with nothing. A barrister is able to assess better than a solicitor.
I hope this helps you make the right decision for you,