Yes it's a bed bug. Your landlord isn't obliged to pay for cost of treatment unless you can prove they were there when you moved in - which it doesn't sound like.
BB are one of the hardest pests to get rid of and it's extremely difficult to get rid of them yourself without a lot of time, effort and stress. Although heat does kill them, it's not hot enough in this country to penetrate all the way through the mattress or other items and maintain that temperature for enough time to kill them and, more importantly, the eggs. Any sprays etc you can buy in shops are not strong enough to kill them as their bodies are good at protecting from chemicals.
I would call in a company now. Do lots of research and make sure it's a reputable one. Where are you based? BB are becoming more resistant to chemicals so you have to be careful, or you can go for a company with uses a different type of treatment. You could contact the council because some offer treatment at a slightly cheaper price than a private company.
If you really can't afford to get a company in (though really I 100% would), get hold of a steam cleaner and thoroughly steam the mattress, take the bed apart and steam in cracks. They also live in cracks of walls and can get anywhere you can fit a piece of paper. You don't need to throw the bed away if you do this - and you risk spreading them if you move the bedding etc to other rooms.
You can get hold of Diatomaceous earth (DE) which kills bed bugs by absorbing the oily, protective layer that covers their exoskeletons. Bed bugs will dehydrate and die within a few hours. Only put a light trail around the skirting boards and bed legs.
Also be aware if you do a chemical treatment you have to sleep in the bed so the BB will come out and walk through the powder/chemicals. Otherwise they can stay in their hiding places for over a year before dying - they're very resilient.
Put all clothes through a hot wash and tumble dry on high heat to kill the eggs. Freezing infected items below -17 degrees will also kill them and their eggs.
However you only need to do this if you have been severely affected and you have lots of items under the bed or in a bedside drawer etc. It sounds like you only have the beginnings of infestation so you should be ok.
If you want to get a sense of how many there might be, lift up the mattress and see if you can see any black dots, like felt time. This is their faeces and will smear with wiping so you can tell. The number of faeces can give a guide to the number of bugs. Also check the folds of the mattress.
Hope this helps.