I'm assuming this is the UK given the mention of "council tax", though I'm not sure where your personal allowance figures come from - the personal allowance for 2015/2016 is £10,600 and for 2016/2017 is £11,000, so (£30,000 - two allowances) would be less than £13,600.
However in any case your predictions are wrong for two reasons.
Firstly, the personal allowance isn't generally transferable between spouses. A small amount of it - 10% - is if one of you pays no tax and the other is a basic rate tax payer, as you would be with an income of £30,000. You need to explicitly apply for this - see https://www.gov.uk/marriage-allowance-guide.
As a sole trader, you would be making the income so would be the one liable for tax. If you were to incorporate and pay your wife dividends from a company, you might be able to make better use of her personal allowance, but you need to very careful about the legal arrangements. Also with a company the tax position gets a lot more complicated and you have to worry about things like corporation tax.
Secondly, if you have deductible expenses, those are subtracted from the taxable amount, not the actual tax due. So for example if you have £15,000 of income all taxable at 20%, your tax bill ignoring expenses is £3,000. If you have expenses of £3,000, then tax is charged on (£15,000 - £3,000) = £12,000 - i.e. the bill ends up being £2,400. Another way of looking at it is that you get 20% of the expenses - £600 - knocked off the bill.