townhouse = terraced house
duplex = semi-detached house
Don't assume that pets will be allowed if it's not stated- it's quite often the opposite, actually. Always ask about pets upfront, if there's any room to negotiate you want to know first thing before you go to see a lovely house that won't allow your dogs.
Average cost of a house- that depends. There are several threads on this forum where people have stated how much they rent for in different parts of the country, so maybe have a search for those. Very generally, outside of London (London rents are an entirely different world), an average rent for a 2 bed terrace would be something like £550-650/month although it can be more or it can be less. Semi detached houses are usually more expensive than terraces.
For two people, budget roughly £200/month for council tax & utilities on top of your rent, depending on whether you want cable/broadband, etc.
There are lots and lots of threads here that address all these questions, btw. Have a read through the housing forum.