The interpetation of time as either "A" (all the physical stuff exists now in a 3-dimensional space, there is no actual representation of a "time axis", and everything moves around and is modified constantly. Time, in this model, is not a feature of the universe itself, but an illusion.
"B" would be that instead, the universe is a 4-dimensional entity (3 space + 1 time dimension) that is totally unchanging, with time being as real as space, nd each imagined slice along the time axis is equivalent, with no specific point along the time axis being labeled as "now". Movement and change (of physical entities within the universe), in this model, is not a feature of the universe itself, but an illusion.
Neither of these has anything to do with the possibility or neccessarity of infinity or periodicity.
If this theory is accurate, then physical reality could potentially have an infinitely extended past, and the notion of an infinite regress of causes becomes a metaphysical possibility.
No, it says nothing about that. Nothing keeps the 4-dimensional "block" that the universe would be in "B" time from having a border in any of the dimensions. The same as the 3 spatial dimensions are not necessarily infinite (nobody knows for sure), the universal "block" could have a sharp edge/surface at the point 0 along the time scale (i.e., Big Bang), or if you take spatial expansion into consideration, the whole affair could be more like a 4-dimensional cone, with the tip being the Big Bang (in fact, if you look at visualizations of the Light Cone, that's exactly what the Light Cone of the event of the Big Bang would be).
Given this framework, is it feasible for the universe to terminate in a state such as "heat death,"
Sure. These time frameworks are just ideas, beyond theoretical, neither of them has any influence on anything we know or assume about the physicality of the universe. One possible scenario is the Big Chill, vividly shown in the speculative video Timelapse of the Future - basically eternal expansion, with everything just eventually slowly disappearing.
Light Cones are a useful tool for physicists to work with their maths, nothing more, nothing less.
with no possibility of rejuvenation?
Again, neither of the two interpretations has any impact on these things. If the universe should somehow contract eventually, instead of expanding unendingly, then who knows what would happen at the Big Crunch.
This idea appears to present a mathematical paradox to me. If time is infinite, it seems that an infinite sequence of events, including infinite big bangs and heat deaths, should have already occurred.
It seems that it's more an (understandable) misinterpretation of the word "infinity". First of all, it is absolutely possible to have infinity only in one direction (positive along the time dimension). Secondly, it is normal for things to converge to a plain, regular number when going to infinity. Thirdly, it is often more helpful to think in terms of "this expansion will never stop" instead of "this will expand infinitely". The latter has the danger of thinking that something that is not infinite may be eventually become infinite - this is a fallacy of course. For example, assuming the universe is infinite in size, you can travel linearly for an unending amount of time, but at each point in time there will be a concrete, finite distance you will be away from your starting point. At no point will you say that you are now suddenly "infinitely far away" from the starting point.
Is it mathematically plausible for an event to occur for the first time after an infinite interval has passed?
Maths would not say that an "infinite interval" has passed. They would say "an arbitrarily large interval" has passed, or "there is no limit to the length of the interval" or something like that. Infinity is not a number. Saying an interval is infinite is a short-hand for saying that it has no limit at one or both of its sides.
Wouldn't this be analogous to an infinitely long rope, where one "half" is blue and the other "half" is a different color—a situation that is mathematically impossible?
By now it should be clear - maths does not talk about physical ropes, and confusing mathematical constructs, especially when infinity is involved, with physical reality, is a way to total confusion. The one has nothing to do with the other. In regards to physical reality, infinity is a shorthand for "it's arbitrarily large and we can't put a limit on it (or there may be no limit)", not something you would directly work with.