I think it largely varies on where you stay and the job centre you visit. My friend was on JSA after giving birth because she had to leave her last job. They kept complaining that applying for between 4 and 6 jobs a week was 'not doing enough' (even though she was doing that whilst looking after her newborn baby)... Yet I've seen other people say they only need to put 1 or 2 down and they were happy with that.

The allowance to work 16 hours or less does apply to volunteer work too, though I think they can sometimes be a bit selective in what sort of volunteer work they class as being acceptable.

My honest opinion, would be to try and get back on ESA. I really do understand that it's a really hard thing having to jump through so many hoops just to prove you're not trying to get something you're not entitled to, but perhaps you could find some form of support service in your area which could help you with it?

I've been on ESA for a few years now, every time they've put me in the work focused group even though I didn't think that was where I should be but I just went to the appointments and kept telling them I hadn't looked for work yet and why. This time however, I was told that if I don't return to work in 6 months (now around 4 and a half) then my benefits will be reduced. I contacted the AS support service in my area and they have taken a look at all the paper work and the 'justifications' the DWP gave for not putting me in the support group. According to them they can pretty much guarantee that I'll be moved to the support group with an appeal which they are currently helping me to put together.

To get into the support group of ESA you have to meet certain key points, a few of these which guarantee you entry are things like;
"inability to deal with small changes, without causing severe stress/anxiety in day to day life",
"inability to engage in social contact due to difficulty relating to others, or to their own significant distress, due to cognitive impairment or mental disorder",
"cognitive impairment leading to daily episodes of aggressive or dis-inhibited behaviour, which are uncontrollable and would be unreasonable in the workplace"

I don't know about yourself, but I match all three of those. Perhaps not every day, but most, and frankly with these things you always tell them about your worst possible day because they are looking for reasons not to give you it instead of looking for reasons to give you it. You know, because the 1.2 billion pound welfare bill is too much, but the 120 billion pound in tax evasion every year is totally fine.