I am wondering how fever causes rise in temperature, or fever might itself be rise in temperature.

Our body temperature rises or falls when we experience fever. But, according to first law of thermodynamics, heat must travel from hotter to colder region, balancing the temperatures.

Why doesn't this happen when we encounter fever?

The temperature from fever goes up for the same reason that your house gets warmer when you turn up the thermostat. We are homeotherms that maintain internal temperature accurately. There is a thermostat. SM

Fever is a "deliberate" act by the body. The idea is essentially to cook the bugs that are causing an infection. The body increases the metabolic rate; more fuel (i.e. food or body fat) is consumed and the temperature goes up.

No, the metabolic rate is NOT increased during a fever (AFAIK). The hypothalmus releases prostaglandin hormones and complicated signalling pathways reduce heat loss from the skin, restricting of blood vessels and cause muscle shivering to retain and generate body heat.

Do you think those muscles don't metabolise stuff?

Do you think those muscles don't metabolise stuff?

Of course they do, but stating it the way you did seems to obscure root cause, as well as ignore tissue specificity.