Kingdom Monera

Reference: NCERT Class XI, pages 13, 14, 16.

Bacteria (Monera) show the most extensive metabolic diversity.
They may be photosynthetic autotrophs, chemosynthetic autotrophs or heterotrophs (saprophytes, parasites, symbionts).
Plants are mostly autotrophic, and fungi and animals are heterotrophic.

Heterotrophic bacteria are the most abundant in nature; most are important decomposers.
They are helpful in making curd from milk, producing antibiotics and fixing nitrogen in legume roots.

Nostoc is a cyanobacterium, a prokaryote; its genetic material is not enclosed by a nuclear membrane.
Penicillium and Agaricus (fungi) and Volvox (alga) are eukaryotes with a nuclear membrane.

Cyanobacteria have chlorophyll a like green plants and are photosynthetic autotrophs; they are also called blue-green algae.
Golden algae are chrysophytes, which are protists.

(a) is correct: methanogens are archaebacteria; they live in marshy areas and in the gut of ruminants and produce methane.
(b) is correct: $Nostoc$ is a filamentous cyanobacterium (blue-green alga) that fixes atmospheric nitrogen in specialised cells called heterocysts.
(c) is wrong: chemosynthetic autotrophic bacteria oxidise inorganic substances such as nitrites, nitrates and ammonia and use the released energy for ATP production; they play a great role in recycling nutrients like nitrogen, phosphorus, iron and sulphur. They do not synthesise cellulose from glucose.
(d) is correct: mycoplasmas completely lack a cell wall, are the smallest living cells known and can survive without oxygen.
Hence the correct statements are (a), (b) and (d).

Hyperthermophiles grow at very high temperatures, and those that also tolerate a pH of about 2 are thermoacidophiles; they live in hot sulphur springs.
Such extreme habitats can be colonised only by prokaryotes.
Most thermoacidophiles are archaebacteria (archaea), e.g., $Sulfolobus$ and $Thermoplasma$; their special cell wall and branched-chain membrane lipids let them survive.
A few eubacteria (some thermophilic bacteria of hot springs) can also grow under such conditions.
Liverworts, mosses, yeasts, diatoms and protists are eukaryotes that cannot survive in such habitats.
Hence the two groups are eubacteria and archaea.

In oxygenic photosynthesis water is the hydrogen donor and oxygen is released: $6CO_2 + 12H_2O \rightarrow C_6H_{12}O_6 + 6H_2O + 6O_2$
Among prokaryotes it occurs only in cyanobacteria, which have chlorophyll a and both photosystems, like green plants.
$Oscillatoria$ is a cyanobacterium (blue-green alga).
$Chlorobium$ (green sulphur bacterium), $Chromatium$ (purple sulphur bacterium) and $Rhodospirillum$ (purple non-sulphur bacterium) carry out anoxygenic photosynthesis: they use $H_2S$ or organic compounds as the hydrogen donor and do not release oxygen.
Hence oxygenic photosynthesis occurs in $Oscillatoria$.

In Whittaker's five-kingdom system, kingdom Monera includes all prokaryotes (organisms without a true nucleus and membrane-bound organelles).
Archaebacteria (archaea), such as methanogens, halophiles and thermoacidophiles, are prokaryotes.
Nitrogen-fixing organisms such as Rhizobium, Azotobacter and cyanobacteria (Nostoc, Anabaena) are also prokaryotes.
So both are classified in kingdom Monera.