We hear of "his mother and sisters"(7), but nothing of his father and brothers. These females have to live somewhere, and on something, and if Mr Elton senior has died, it would be quite natural for them to live in the family home, and this could have been the 'independent property' Mr Elton inherited - possibly with a jointure to prevent him turfing his mother out.

One of the advantages of a church living is that you are given a house to live in while you hold it. (and even the yellow curtains and the whitewash might have been paid for out of the dilapidations of the previous incumbent)

He might have been economizing on the carriage by choice, and merely expressing his disappointment to his bride that the wedding would need to be delayed a week due to a dilatory coach-maker, but if he could afford a carriage without monetary assistance from her, why would the wedding be put off for the purchase of a coach? Captain Wentworth was rich enough to afford to purchase a Landaulette after he married.
Mrs Elton feel the need to pull herself up with a slight cough when she mentions the subject, and I think it might be because the coach had been slipped in as part of the cost of the wedding, to be paid for by the family of the bride, and when uncle Hawkins had identified this unnecessary but expensive inclusion, he might had suggested it should rightly be purchased by Mr Elton as part of his married establishment, and Mr Elton had therefore needed to suggest to his bride to be that without a degree of co-operation from her family about the wedding arrangements, they might never be married at all. Better that than throwing himself away.