The speed of light is an upper limit on how fast matter, energy, and information can move through space. The expansion of space during the Big Bang (and afterwards at a much slower rate) is something fundamentally different from motion. It’s a “metric expansion,” which doesn’t require anything to move within space and is not limited by the speed of light. It’s not even measured in the same units as speed; speed is distance/time, whereas expansion is (distance/time)/distance, which oddly collapses down to just units of 1/time.

yes and no, there is actually a specified timeline of the universe's expansion down to the minute and even the second(or less than second lol). let me see if i can find the one that I was taught on... ok, so on second thought... Obviously I can't just upload a book to reddit, but the graphic and explanation in this link is similar enough to my understanding that i'm trusting it to teach you. https://www.physicsoftheuniverse.com/topics_bigbang_timeline.html

basically, it cannot be said that the universe expanded at the speed of light for multiple reasons:

one being the universe didn't expand, it is currently expanding. It never stopped, we can measure it right now if we wanted to and if we were rich enough. I had to specify that first because if I didn't I would be allowing you to be misled.

two, being it entirely depends on which distance away from us (or Planck's constant, which was the very beginning of the very beginning, meaning it is currently impossible for us to measure anything before it occurred because it was the beginning of time as we know it) and at what point in time you are measuring. If we measured a galaxy next to us right now, it would not be expanding away from us at the speed of light because the distance between us is way smaller, but if we were to measure a galaxy on the other-side of the universe it would most likely be expanding away from us past the speed of light because the distance is *impossible to comprehend*. Keep in mind speed is just distance/time, and so is unfortunately not very helpful in this specific example... i'm actually pretty unwell rn so if this explanation makes absolutely zero sense, I sincerely apologize, I tried my best and if you have further questions I will try my best again to clarify.