The Combine in HL are the Precursors, the Advisers are young Precursors who have self evolved themselves into the simplified forms they are now. We know the Combine have extra dimensional travel and occupy multiple parallel universes already, so it makes perfect sense that they can occupy the ‘Halo’ universe as well as the HL universe. The Xen lifeforms are like the Flood in that they were biologically engineered by the Precursors. Unlike the Flood, the Xen lifeforms were discarded as failures and placed together on the Xen home world until the Resonance Cascade occurred, teleporting a large amount of them to Earth where they helped break down human society as the Combine quickly assumed control. Gordon destroyed the Nihilanth, the overlord of Xen, allowing the Combine to tame their creations once again. The G-Man, having access to Precursor technology, uses his seemingly nigh-omnipotent powers to travel constantly between universes to make sure all the events go according to plan. For some strange reason, the Precursors are bent on making humans the holders of the mantle in the ‘Halo’ universe, while keeping them down in the ‘Half-life’ universe. Eventually, G-man meets with Master Chief for the first time, just as he did with Gordon and Shepard, and gives him similarly vague, ominous messages about certain events that are about to unfold.
What this accumulates in is the Precursors returning to their original universe using the human experimental technology of the Half-Life universe. The rest of Earth is caught in the effect however, and brought along. Certain unforeseen consequences may involve the re-emergence of the Flood, which may combine with the Xen lifeforms that are brought in by Precursors (the Flood fuse with the headcrabs to create a new kind of parasite) and a cosmic moral and political dilemma involving who is the enemy and who is ally and what the Precursors’ true initiative is for their seemingly contradictory history and philosophies.
Update:
This is what happens. Well, long ago, when the Forerunners apparently vanquished the Precursors, in reality, using their tier 0 technology the remaining Precursors fled to an alternate dimension where physics and time differ from the ‘Haloverse’. There, they began the development of the ultimate bioweapon, which would eventually take the form of the Flood infection. However, the failed experiments were contained to a floating cavernous mass of rocks, which would come to be known as the Xen homeworld, the headcrabs were the last failed experiment before the invention of the Flood, explaining their similar parasitic nature. A compound designed to spark the Floods creation was transported back into the Precursor’s home dimension, which it was discovered by humans and eventually evolved into what we know now as the Flood menace. However, the Precursors did not plan to devastate the universe with their weapon, instead they had a much more mysterious motive. Without enough power left to return the entirety of their species to their home dimension, the Precursors sought an alternative gateway. In an alternate dimension, where Earth exists much as it does in the “Haloverse” but at a different time, Gordon triggers the Resonance Cascade, allowing the lifeforms of Xen as well as the Precursors to enter their dimension. The Precursors, at this point self-evolved into giant larva like beings, begin to quickly take over Earth and convert humanity into a makeshift army of soldiers, their highly superior intelligence and technology making it easy to manipulate and use their Xen bioweapons as well as the primitive humans. A small resistance containing Gordon Freeman and others such as Alyx Vance somehow manages to interfere however. The Combine’s interest lies in an experimental teleportation technology developed by Aperture Science under the code name “Borealis” which they believe will allow them to return to their home dimension where the UNSC and the Covenant exists and complete their mysterious objective. When they activate this technology, Gordon Freeman and other members of the human resistance get caught in the effect, and enter this new dimension with them and a large amount of Xen lifeforms. The Precursors reawaken the Flood, which absorb the headcrabs and fuse their DNA together, creating even more powerful Floodcrab infection forms that create incredibly strong combat forms. Gordon and Master Chief become the main characters in this new battle, which is fought between a giant variety of alien creatures. Many Voritgaunts are enslaved by the Covenant and used in the war.
Also, G-man is a human from yet another alternate dimension, who has been recruited by the Precursors for some mysterious reason and given reality altering powers as well as the legendary item ‘the Organon’ which is what he carries in his briefcase. He sees to it that Gordon Freeman and other essential individuals to their goal are kept in line and don’t stray from their predictions. Ultimately, when everything is in place, G-man is tasked with activating the Organon, which will activate all Precursor artifacts in the universe (similar to the Halo effect), in turn accomplishing what the Precursors/Combine secretly planned for billions of years. Creating God.