By all means kill the little bastard - if it is infected with a pathogen, letting it go will only let it potentially transmit to another poor soul.
Speaking of the pain of the bite, When I feed mosquitoes on my arm at work (i’m a medical entomologist), mosquites that have been colonized in the lab for some time hurt a lot more than wild mosquitoes. In the wild, there can be fairly heavy selection for less painful bites - less painful equals less host defenses and a greater chance for the mozzie to get away with it’s booty (your hard-earned blood). Mozzies that have been colonized in the lab aren’t under any selection to keep painless bites - they get fed anyway. If you feed mosquitoes on a consistant basis in the lab, you will become desensitized to bites from that species and will not react, although other species will still illicit a response.
Some mosquitoes can lay an initial batch of eggs without imbibing a blood-meal (this is called autogeny). If they develop in good conditions as larvae, they have energy reserves stored up when they emerge as adults and can oviposit a small batch of eggs. There is variation among mosquito species as to their ability to lay autogenous eggs, and in species that can do it, there is often variation among individuals in the ability. There are even some mosquitoes that never take a bloodmeal. They belong to the genus Toxorhynchynes, and the adults (both male and female) live on plant sugars, while the larvae are predacious on the larvae of other mosquito species (they are sometimes used as a biocontrol agent for mosquitoes in certain circumstances).
Hope that helps 
MM