This isn't a bad question, but there's a subtle problem with it I'll have to address to give an answer, which is that there's no such thing as "the" BBT. The BB as a historical event doesn't have (as much) predictive power before we add theoretical details that have varied over time. So even if we answer your three part-questions, what comes next will likely be thought of as "a" BBT once its details are known.
Since "the BB happened" was first proposed, we've added a positive cosmological constant, cold dark matter and, depending on how you define "the consensus model", early-universe inflation. These each had empirical motivations, and if you'd asked your question a few decades ago the answers to its parts would have been something like:
1. It can't explain flatness, homogeneity, and the very low magnetic monopole density required given that they're unobserved. (The $\Lambda > 0$ and CDM details are empirically recent.)
2. The maths says something else if you assume the time-dependence of the scale factor that follows from the then-expected historical profile of what kind of matter dominates.
3. That question was probably hard to answer at the time, but in hindsight we needed the effective early value of $w$ to be negative enough, certainly $w<-\frac{1}{3}$ and ideally $0\le w+1\ll 1$.
Today, most physicists consider inflationary cosmology the most promising approach to these. 484
Assuming they're right, today we have very some different problems to worry about, and our desire to do something new has theoretical motivations. These are problems of quantum gravity, which unfortunately also have a bearing on how we should model inflation.
Traditionally Big Bang cosmology has been an exercise in general relativity, but we know gravity's behaviour on small length sales will require something new, and the way the Big Bang worked very early on will depend on how quantum gravity works. So:
1. We don't know what happened in the "first" <PERSON> time or so; it's not even known whether anything existed "before" the Big Bang. Although <PERSON> has popularised the analogy that this question might be as wrong-headed as asking what's North of the North Pole, he's also contributed to a model in which a pre-BB era occurs.
2. We don't know nearly enough about quantum gravity to settle the details of the <PERSON> epoch. (In fact, this might remain a purely theoretical, not empirical, question indefinitely.) For example, string theory's predictions differ from those of loop quantum gravity, or (insert your favourite even less popular alternative here; see this list). There are reasons to favour string theory, to be sure; but they are theoretical reasons that attract a lot of controversy, not empirical reasons that attract a lot of consensus.
3. Explaining the needs of quantum gravity would probably take far too long for an adequate answer here, so I suggest you begin learning about this topic here, then choose what to read next based on which sub-topics take your initial interest, your mathematical expertise etc. 651