We use a dual network card server with bond in 802.3ad mode.
While LACP was first introduced in IEEE 802.3ad, it's since been moved to IEEE 802.1AX.
So I have questions about how the ARP protocol works in bond mode and why it affects the distribution of incoming bandwidth.
LAG combines multiple physical ports into a single logical port. It needs to use some form of traffic distribution. How that distribution works on either end is not part of the standard, however - see 802.1AX Clause 6.2.4 Frame Distributor:
This standard does not mandate any particular distribution algorithm(s)
Most implementations use a physical port index derived from a hash function over any subset of {source|destination} {MAC address|IP address|transport layer protocol|transport layer port number}.
Since you're particularly asking about host ingress distribution, that part depends on the uplink switch, its implementation and configuration. Most often, it's best to use a hash over IP addresses and L4 port numbers.
In the case of dual card machine, will these two network cards exposed as a single IP and MAC addresses when in bond and 802.3ad mode?
There are multiple bonding modes, but in LACP mode, all NICs use a single MAC address and as many IP addresses as you've configured on top.
If so, how will the ARP protocol work and which network card will respond to ARP request?
The ARP protocol works completely normally over the single logical interface. Which physical interface sends the ARP responds (or any other frame) doesn't actually matter because its MAC address is only associated with the switch's logical interface. Traffic distribution works as outlined above.
That said, LACP might not be the best bonding mode for your use cases. You should also look into using separate NICs (with their own MAC and IP addresses) with a round-robin access method on the application layer, e.g. via DNS. Or simply a faster NIC.
