The reason a bank wants you to have 'equity' in your house (meaning, value that you hold free and clear, without anyone else being owed money for it) is that it makes it harder for you to ignore your obligations and run away. ie: if you own a house worth $200k, and owe the bank $150k, you wouldn't stop paying your mortgage and move to Venezuela, because you would be losing the net $50k equity that you have in the house.
Having equity in the house also helps when/if the value of property declines. In the above example, if house prices dropped 10%, your house would be worth $180k, but you would still only owe the bank $150k, and would thus still have 'equity' in the house. This again makes it less likely you want to run away from your obligations, meaning the bank will continue to get their mortgage payments.
With that in mind, consider your questions:
Your cousin will, at the end of all the paperwork, own a house worth market price, on which they will owe the bank only a fraction of half the value of. ie: if the value of the house determined by the appraiser is $100k, then they will owe the bank only for the amount of money needed to by out their sibling's $50k ownership portion. If the will says, for example, that they must pay 90% of the appraisal value for that remaining portion, then they would end up taking a mortgage of $45k, and would own a house worth $100k (this means they would have 'equity' of 55k).
There are limits on how much a bank will lend compared with your equity %. Some limits are defined by your national government, and some are limits that the bank imposes on itself to limit its risk. Typically, having between 5-20% equity in the house is considered good / 'good enough'. This implies that with a reasonable credit rating, your cousin would be very likely to get a mortgage from the bank.