Should I be paying UK tax on my US investments?
As regards the unremitted US income the technical question to ask is .... are you within the terms of Section 809D ....... Application of remittance basis without claim where unremitted foreign income and gains under £2,000? Section 809D was inserted into UK tax law by the Finance Act 2008.
That is, the amount of the dividends from the US mutual fund, plus your share of the interest on the US bank account .... is the amount of income less than £2000 per year? If the answer to that is "yes" then effectively the remittance basis still continues for you .... the income is not taxable in the UK unless you remit that income to the UK.
And what on earth is non-dom and how do I know what I am?
There are three characteristics in UK tax law, that assist in determining tax liability on income or gains ..... Resident .... Ordinarily Resident .... and Domicile.
Based upon what you have posted it looks likely that the UK tax authorities would treat you as .... Resident .... Ordinarily Resident ... but Non-Domiciled! Accordingly, if you are within the terms of Section 809D mentioned above, specifically the £2000 limit, you are not liable to UK tax on overseas investment income not remitted to the UK.
Compare that to me ... Resident ... Ordinarily Resident .... Domiciled in the UK ..... if I happened to have your US income, I would be liable to UK tax on the income even if the income is not remitted to the UK.
Does that help?