(A) What does the mean two router connected by PPP(point to protocol)? As per as my understanding two routers are directly connected.
Routers (or any nodes) may be connected using PPP. PPP uses an underlying simple serial interface like RS-232 and provides data link layer functionality required by IP (data framing).
(B) Why if the two routers connected by PPP then ARP doesn't required?
These interfaces are point-to-point and use no addressing, so there's no use for ARP.
For completeness: The PPP framing specification (RFC 1662) does include an 8-bit address field, but since L2 addressing is not used (the framing is borrowed from HDLC), that address field is specified as a constant 0xff:
3.1. Frame Format
The Address field is a single octet, which contains the binary
sequence 11111111 (hexadecimal 0xff), the All-Stations address.
Individual station addresses are not assigned. The All-Stations
address MUST always be recognized and received.
Accordingly, a PPP frame with any other address is invalid.
(C) What does the mean that the two routers are connected via ethernet?
Routers may be connected via Ethernet with a point-to-point link (simple cable) or using switches. In any case, Ethernet is a point-to-multipoint network, requiring proper MAC addressing.
(D) Why if the two routers connected via ethernet then ARP is required?
ARP is only used for IPv4 over MAC-based networks, most prominently Ethernet/IEEE 802.3 or IEEE 802.11.
MAC-based networks require the protocol on top to specify the desired destination MAC address for their payload. IPv4 uses ARP with a broadcast address to learn that destination MAC. Then it uses the underlying network as L2 to deliver its packet.
You cannot run PPP on top of such a point-to-multipoint network, but there may be special variants like PPPoE.
PPPoE creates another L2 on top of Ethernet's L2, but there's no addressing between PPPoE and IP still (PPPoE uses MAC addressing downwards with Ethernet, ie. Ethernet's MAC addressing is controlled by PPPoE, not IP).
(E) If the two routers are directly connected then they can be communicated via IP address or Mac address or both?
Any nodes using IP use underlying data link layer (L2) protocols for local communication. The L2 protocol can vary along an IP path. Ethernet includes the required L2 protocol, a serial link requires PPP, (obsolete) SLIP or something similar.
Basically, routers communicate at the network layer (L3), using IP addresses. As pointed out above, they also need to use proper L2 addressing on certain networks, hence the need for ARP (or ipV6's NDP) as 'glue'.