There is no need for a solicitor to be involved in probate and administering an estate. The first step after your father dies (and after registering the death and arranging the funeral) is to ascertain all his assets and liabilities. This gives you a net value of the estate. If the net value is over �312,000, there may be IHT to pay (if no spouse or charitable exemption) and thus you need to pay IHT first. Then you make an appointment with the probate registry and go there for a short interview and they then issue you the grant (if there is no IHT, skip the "pay IHT step" and go straight to the Registry). Just one of you could apply for the Grant if you wished with "power reserved" to the other (useful if one lives abroad). Once you have got the Grant, you collect in all the assets and pay off all the debts. It is useful and quite normal for joint executors who have extracted the Grant together to set up a joint bank account to do this. Up to you how you set up the signature procedures, but bear in mind that BOTH of the executors are jointly and severably liable for the default of the other (ie, if one of you pays money out that shouldn't be paid, both of you are personally liable, so probably best for you both to sign if there is any suspicion that one may not be strictly trustworthy). As for selling the house, you might want a conveyancing practitioner to deal with the formalities and pay you the funds.
If the estate is large or complicated, you may wish to take initial advice on potential tax saving implications, otherwise, no reason why you can't do it all yourself.