DNA replication

DNA-dependent DNA polymerase catalyses polymerisation only in the $5'\rightarrow3'$ direction. Hence replication is continuous on the template strand with polarity $3'\rightarrow5'$ and discontinuous on the template strand with polarity $5'\rightarrow3'$. Both statements are correct.

Matthew Meselson and Franklin Stahl (1958) proved that DNA replicates semiconservatively.
They grew $E.\ coli$ for many generations in a medium containing $^{15}NH_4Cl$ as the only nitrogen source, so that its DNA became heavy.
The cells were then transferred to a normal $^{14}NH_4Cl$ medium and DNA was extracted at definite intervals and centrifuged in a CsCl density gradient.
After one generation (20 minutes) the DNA had an intermediate (hybrid) density; after two generations (40 minutes) equal amounts of hybrid and light DNA were found.
This showed that each new DNA molecule has one parental and one newly synthesised strand.
Hence it was first demonstrated in $Escherichia\ coli$.

DNA polymerase can add nucleotides only in the $5' \rightarrow 3'$ direction.
The two template strands are antiparallel. On the template with $3' \rightarrow 5'$ polarity, synthesis is continuous (leading strand).
On the other template, with $5' \rightarrow 3'$ polarity, the new strand cannot grow continuously towards the fork. It is synthesised as short pieces called Okazaki fragments, each made in the $5' \rightarrow 3'$ direction.
These fragments are later joined by the enzyme DNA ligase after the RNA primers are replaced by deoxyribonucleotides; together they form the lagging strand.
Each fragment grows $5' \rightarrow 3'$, but the lagging strand as a whole appears to grow in the $3' \rightarrow 5'$ direction. In this way the Okazaki fragments explain the apparent $3'$-to-$5'$ DNA replication.
The semi-conservative nature of replication was proved by Meselson and Stahl, and Okazaki fragments have nothing to do with transcription.
So Okazaki fragments polymerize in the $5'$-to-$3'$ direction and explain $3'$-to-$5'$ DNA replication.

The bacterial chromosome is a single circular DNA molecule with one origin of replication (ori).
At the origin, the DNA double helix opens to form a replication bubble with two replication forks.
The two forks move away from the origin in opposite directions around the circle, so replication proceeds in both directions (bidirectional replication) until they meet at the terminus. This gives the theta structure seen by Cairns.
Telomerase is found only in eukaryotes, which have linear chromosomes with telomeres; it is not needed for circular bacterial DNA.
RNA primers are indeed needed to start each new DNA strand, but the feature that describes how synthesis proceeds from the origin, as the question asks, is bidirectional movement.
So DNA synthesis starts at the origin and moves in a bi-directional way.

In semiconservative replication, the two strands of the DNA double helix separate, and each acts as a template for a new complementary strand.
So each daughter DNA has one old strand and one new strand. This was proved by Meselson and Stahl (1958) using $^{15}N$.
So the method is semiconservative.