In the UK, is it possible to be in full time employment and be a sole trader at the same time?
i.e. Work a full time job as an employee. Do contract work as a sole trader on evenings and weekends.
Similarly, is it possible to run a private limited company and be in full time employment and/or sole trading at the same time?
There is no legal reason why you cannot be employed and self employed.
You will need to register as self employed and, I would suggest, notify your current employer though that is not a legal requirement outside of anything in your employment contract.
You will need to do self assessment each year and you will also need to start paying Class 2 (currently only £2.70 per week https://www.gov.uk/national-insurance/overview ) and possibly Class 4 National Insurance contributions on your profits as well (% based).
There is some good information about tax and NI for someone who is employed and self employed on the HMRC website http://www.hmrc.gov.uk/working/intro/employed-selfemployed.htm