And, theoretically, if it were successful I would then receive recourse to public funds due to having ILR. Is that right?
If you are granted ILR you could claim benefits, but it isn't much anymore as that is ending. It will be a fixed amount of money for rent, which likely won't cover all the rent of your room as they only pay at the 30th percentile of rents in that area. If you are under age 35, it will will be the room rate and over 35 will be the 1 bedroom rate. Plus about £74 per week Jobseeker's Allowance, but you would need to use that £74 to top up any rent shortage, pay your Council Tax, pay for heating, water, food and clothes. Not paying your rent and then getting evicted, is seen as intentionally making yourself homeless and then there is then no help from your council as they have already given you some rent money.
However, all the income based benefits are being replaced by the One Income Based welfare payment called Universal Credit and that is being run by the DWP. It is likely that UC already in in your area for new new claims and UC will be in all areas by the end of next year for exisiting claims too. About 70 DWP job centre areas are already going onto UC every month from 2017.
For Universal Credit, the UC conditions are that you would be required to job hunt for 35 hours a week, visit the job centre, go on any work experience, go to a work provider, apply for any jobs they tell you to etc. There are tough sanctions too (loss of benefits): 3 months for the first sanction and something like 3 years for a second sanction.
The only way to avoid the Universal Credit these conditions, is to earn at least 35 hours a week, every week, at the hourly national minimum wage. Parents too will soon have to meet these new welfare requirements of earning a minimum amount each week, to avoid the Universal Credit conditions mentioned above.
As you are in a women's refuge, ask them about asking for Public Funds because the staff there they will know if you are likely to be allowed to get that and how little that amount will be while you await to see if you can have ILR on DV.
If I apply for ILR based on DV and am rejected, what happens? Is my spousal visa cancelled immediately and I have to leave the country? Because I've just told UKVI that I no longer meet the conditions of the spousal visa.
UKVI will give you a date to leave the UK by. The Immigration Act 2016 ended all in-country appeals. They might let you appeal from inside the country, but they don't have to. I assume you could ask for a Mandatory Reconsideration, where they look to see if they (UKVI) have made a mistake when they refused you ILR.
Constant abuse just gets things shut down and these people will always make it harder for genuine claimants too, and not just for immigration. i.e. the going to their doctor and claiming "stressed and depressed" to get sick benefits, that became so popular on "UK benefits' forums. That then brought in the 2008 change of, medicals must now be taken for sick benefits; the reduction in their sick benefits to the same rate as the jobseekers; The MAC reporting what UK benefits are popular with foreign natioanls/those who were a foreign national when they first applied for a NINo and those benefits ending or being changed; etc