Request help to identify insects or other creatures. Please post the location that the insect you want to identify came from, this will help greatly in species determination.
Yes, it does seem to have 8 legs, and I am almost certain it is not a bed bug. The photos do not look like one of those.
Whether it is a mite or some sort of spider (you mentioned 'one of those small red spiders') I cannot tell, and certainly am not knowledgeable about either of those arthropods.
It would help if you explain a bit more, particularly where it was found (presumably the US since you chose that flag, but you could be an American overseas somewhere), and also where it was found - in a bed or somewhere else in the house?
Did you notice any particular behaviour? Was it fast or slow moving and did you notice anything else?