How do you even qualify for SH?
You can get an application form from your local council, fill it in and send it back, they'll assess your needs and priority level. Many councils now operate a choice-based lettings system where they advertise empty properties each week and you register your interest on the ones you would consider accepting (called bidding). As well as allocating a priority level, my local council also tells you what types of property you are eligible to "bid" for so if you were a single person you would not be allowed to bid for 2-3 bed properties, people with children under 10 aren't allowed to bid for flats higher than the second floor, to bid on bungalows one of you must be aged over 55 and/or have a qualifying medical condition, and so on.
How exactly does the SH system work then in terms of paying rent?
The same way a private letting works. Rent is £xxx and due on a specific day each week/month. The tenant either pays the full amount of rent themselves or - if they qualify for it - they claim Housing Benefit for some/all of it. Many tenants in social housing work and pay full rent, the situation in the South East is not indicative of the entire UK and in lots of areas there is little difference in rent levels between social housing and the private sector, there is also not the same social housing shortage.
What would happen if say a couple were both unemployed, no intention of finding work and then decided to have more children?
Depending on how many children they have they may become overcrowded. Councils have rules on how many people can occupy a property according to how many bedrooms it has so if they exceeded this they could apply to rejoin the housing register and start looking for another property. They wouldn't get one instantly as they'd have to wait for a suitable property to become available and may not be first in line for it when one does become available.
Would this effect others in SH?
It would have no affect on existing tenants and even after they were moved the family would only have one property, their old property would be let to someone else. For example, Family A is in a two bed with three children. My local council would class this as overcrowded and award a Band 2 priority level for it (they have five bandings - Priority, then Bands One through Four with Four being lowest priority). They register a "bid" on a three bed property nearby and are successful. They move in. Their two bed property is now empty and is readvertised, it is let to a couple with two children.