a) Yes, starting with Aristotle.
b & c) One of the earliest philosophers who considered that time had a beginning is Aristotle.
This was later picked up by Islamic philosophy with Ibn Sina (Avicenna) as being the most famous exponent. In fact he considered time as such a fundamental category that he temporalised logic.
Then Avicenna's thoughts on time and it's beginning was incorporated into Christian thought by Aquinas.
Once Einstein came up with GR he looked at what it had to say about cosmology. At first he assumed that the universe was always there and static - so he assumed a universe without a beginning in time. However, GR showed that the universe was dynamic. But it was both Friedmann and Lemaitre that actually showed that the universe was expanding but only Lemaitre that took the further step of stating the universe began in a 'primeval atom'. These theories were more or less rejected out of hand by Einstein but astronomical observations by Hubble showed that the universe was in fact expanding and Einstein very quickly reversed his position endorsing both Friedmann and Lemaitre.
More recently, Lawrence Krauss has written a book, A Universe from Nothing which is, as David Albert pointed out, is philosophically incoherent. A universe cannot begin from nothing. The vacuum is not nothing and nor are physical laws. Nevertheless, he is not denying the universe had a beginning.
Lee Smolin has a theory of a Darwinian multiverse theory where universes have beginnings but the multiverse does not. However it should be emphasised that this, presently, has the status of a hypothesis and that it is difficult to concieve exactly how such a theory can be empirically tested.
d) Given that Hume rejected Christianity, I would suggest that it's very likely he disputed that time had a beginning given how deeply embedded that notiin is in Christian philosophy. And in fact, he has Cleanthes in his book Dialogues concerning Natural Religion disputing exactly this point. It's likely that Hume neither believed that time had a beginning and nor time is eternal. He simply considered it not proven. (It's not well known that Hume influenced Einstein in thinking through the notion of simultaneity in his special relativity. This appears to be due to his scepticism about how one moment of time succeeds another. Of course, time has a rebirth in QM as a reality rather than as an aspect of geometry. This is a line that Smolin is pursuing currently).
