You can elect to be taxed on a cash basis, due to rules commencing in 2013-14. If you do not elect for the cash basis, then you can claim capital allowances for equipment, but the beauty of that is that the capital allowance claim can be deferred, to a future year when you may not have surplus personal allowances and it makes a real tax effect.
If you bought the computer for the purpose of the business, then you can bring in the entire cost of the computer, even if purchased a bit before the commencement of trade. If you did not buy it for the business, but happened to have it and make use of it when the business started, then you have to restrict the claim based on its market value on commencement of trade. As you can imagine, second hand computers do not have much by way of value. Finally, if and when you make a claim for the cost of the computer, you would have to restrict the claim proportionately for any private use. If not claimed in the first year, you would have to limit the annual claim to 18% of unclaimed residue each year (before private adjustment), and if you dispose of the machine you would have to account for its value as a receipt.
As you say, probably not worth the hassle, especially if you are restricted to market value at commencement. And if you claim for cash basis to apply the whole thing becomes otiose.
As for household running expenses, you may find it worthwhile checking the fixed scale rates provided under "simplified expenses" - HMRC website gives details. However if you have no other taxable income there seems to be no benefit in claiming, at least for that year.
Strictly, you don't get a choice not to claim expenses that are proper to the business. Although clearly not relevant to you, there is a common VAT fraud in which taxpayers suppress their sales to avoid registration and also suppress their expenses to make the margins look right. The profit of the business is a matter of fact, although in practice there are no direct tax penalties for overstatement (unless it gives rise to an understatement in another year).
Basically, stop worrying about it. Just fill in the figures that you mentioned and all will be well. Are you doing 30+ hours per week in this? Claiming working tax credits (while they are still around)?