Why have bacteria not evolved to fill the extreme environments to the same extent as archaea? What has enabled archaea to colonize these areas when bacteria do not?
I suspect this falls under the no-homework policy, so I will not give a full answer, just lots of links to point in the right direction. A good reading material is the genome sequence papers, such as that for Methanopyrus kalderi.
The domain Archaea contains some of the most thermophilic organisms of this planet. Very thermophilic organisms are called hypethermophiles, but employ very much the same strategies.
However, not all thermophiles are archaea, some notable bacteria phyla that are thermophilic are Thermotogae, Aquificae, Thermus-Deinococcus clade and some phyla have several thermophilic genera, such a Firmicutes with Geobacillus. Taq polymerase for PCR comes from Thermus aquaticus (its relative Deinococcus radidurans was nicknamed Conan the bacterium due to its radiation resistance, so is notable for other reasons), Thermotoga maritima was a subject of a structural genomics screen and is highly studied and Geobacillus is a treasure trove for biotech (see PubMed).
These bacteria share many genes with archaea by HGT.Thermotoga maritima shares 24% of its genes with archaea. So it is best to talk about thermophilicity in general.
Not all archaea are thermophilic. The methanogens, such as Methanobrevibacter, are mesophilic and actually a lot of research, such as vaccine and methane pathway antimetabolite development is ongoing.
Firstly, the cells do not need to burst. To do so they employ a shield of protein that pack in a crystalline lattice called the S-layer. Geobacillus a firmicute (i.e. monoderm with thick peptidoglycan) has it too despite the peptidoglycan.
Firmicute spores are annoyingly resistant as they contain dipicolinate. Geobacillus expecially so.
Secondly, the DNA must not denature. Archaea have high GC content, but the thermophilic bacteria generally don't, hence why biotech likes the latter more. A key feature is high levels of polyamines, which stabilise the phosphate groups.
Archaea, but not bacteria, have a unique membrane composition, which is reviewed here.
Every protein has a melting point. The more contact it has the more stable it gets. These often require expensive to make amino acids. There are many analyses of amino acid composition of thermophiles in PubMed, but be warned that some papers suffer from bias from archaeal high GC.
Now, a thing to note is that an thermophile will do poorly at a lower one. This is because of enzyme kinetics and temperature dependence. Namely, the kcat and (less so) the kon and koff of the MM change with the Arrhenius equation in the transition state theory. In a more accurate model, it follows the Eyring equation. In actuality is even more complicated (and fascinating) and explained with the Macromolecular rate theory (and here) where the fluidity of the 3D structure of the protein plays a part. But in short, turnover (k_cat) and Michaelis constant (K_M) go up with temperature, while the former is good (faster), the latter is bad as it needs to be close to physiological substrate concentration. Going down in temperature, the enzyme will be slower, but will find its substrate.