The cash gift from his father is a side issue. In your circumstances, I think he should offer to share it but I don't suppose that he is obligated to do so.
However, his position with regards employment/benefits/signing on is intolerable. And very unwise from a financial perspective- most especially in the long-term/looking at pension. He may not get much in the way of benefits due to your income but if he has little or no private/occupational pension then he will really need to sign on to get his contributions paid for him so that he can maximise his state pension entitlement. Unless he's planning on living off your pension? If that is the case then I hope you have substantial provision or finances may be very tight.
In the short-medium term- I think it's a somewhat disrespectful that he is happy for you to work full time whilst also taking on additional tutoring to make ends meet whilst he doesn't work and, more importantly, it sounds like he makes only sporadic efforts to find employment.
I suppose the most important question is whether you are happy with this situation? Personally, I would be quite resentful of the fact that he hasn't worked in 3 years, won't sign on and only sporadically looks for work whilst being happy to see me work full time and doing additional work. I would be saying he needs to be taking on some work- even if it is boring/unskilled work that he'd rather not do. I'm not saying he'd walk into a job tomorrow without any effort, but I think if he tried hard enough he would find something within a reasonable time-frame even if it wasn't his ideal job.
I think that in such circumstances, i.e. you taking on extra work because he is bringing in no money (and is not a SAHD or a carer), cut backs would need to be made to allow me cut back to just working in my usual job. For example, running 2 cars on one income sounds something of a luxury that could be cut back on- you need yours for work, so it is would be his that would have to go. Maybe when it's not so nice for him, but it might allow you to cut back your hours.
Assuming he is in reasonable health, I also don't think I'd happily accept that a man in his mid-50's "will never work again". He can and should be looking for work- there are only 3 reasons why a person of your DH's age should not continue to look for pain employment: 1)They are unable to work due to illness or disability; 2)you have sufficient savings/income that you both can afford to take early retirement (in that scenario it would be ok for one spouse to take early retirement and the other to CHOOSE to continue to work); 3) if he was a SAHD or a carer for someone. From your OP it doesn't seem as though any apply to your DH.
In short, unless your DH has a good reason for not working, then he (illness/disability/SAHD/carer) needs to do his but to support the family finances. Or he'd be managing his own finances as a single person, I'm afraid. I think I'd feel used- it's not the fact he doesn't work per se, more the fact that he isn't desperately looking for work and is quite happy for you to have 2 jobs.