You inherited your mother's half of the property and the "cost" of that for CGT purposes is whatever it was worth when your mother passed away. If you buy the other half of the property, whatever you pay for it will be added to the "cost" of the half you inherited to give you the full cost of the property for CGT purposes. You can make gains of £11,100 tax free in any one tax year. When you inherit assets, simply because you didn't pay for them does not mean they have zero cost for CGT purposes. Using the figures you provided, you would make a gain of £20,000 of which the first £11,100 would be tax free.
Rental income has always been taxable so I don't know what you read which suggested otherwise. If the property stayed in your name, all the rental income would be yours for tax purposes. If the property was in joint names, it would be split on a 50:50 basis unless it was actually owned in proportions other than 50:50 and you informed HMRC on a form 17. Whose name the mortgage is in is irrelevant as far as the ownership is concerned, though the lender may want your husband as a part owner as security.
I cannot advise you how to buy the half of the property you don't own as I'm not allowed to do so.
I hope this helps but let me know if you have any further questions.
Your understanding is correct. If the probate value of the half of property you inherited from your Mother is £50,000, that is the cost of that half for CGT purposes. Add in the cost of the other half at say £50,000 and your total cost is £100,000. As you say, only the growth in value from when your Mother died is taxed in addition to the growth in value of the second half over what you paid for it. You cannot claim the mortgage redemption fee I'm afraid. There is a tax case on the subject here.