Let's start with the IO scheduler first. There's a IO scheduler per block device. Its job is to schedule (order) the requests that pile up in the device queue. There are three different algorithms currently shipped in the linux kernel: deadline, noop and cfq. cfq is the default, and according to its doc:

The CFQ I/O scheduler tries to distribute bandwidth equally among all processes in the system. It should provide a fair and low latency working environment, suitable for both desktop and server systems

You can configure which scheduler governs which device via the scheduler file corresponding to your block device under /sys/ (You can issue the following command to find it: find /sys | grep queue/scheduler).

What that short description doesn't say is that cfq is the only that looks at the ioprio of a process. ioprio is a setting that you can assign to the process, and the algorithm will take that into account when choosing a request before another. ioprio can be set via the ionice utility.

Then, there's the scheduler. Its job is to allocate the CPUs amongst the processes that are ready to run. It takes into account things like the priority, the class and the niceness of a give process, as well as how long that process has run and other heuristics.

Now, to your questions:

What is relationship between IO scheduler and CPU scheduler?

Not much, besides the name. They schedule different shared resources. The first one orders the requests going to the disks, and the second one schedules the 'requests' (you can view a process as requesting CPU time to be able to run) to the CPU.

CPU scheduling happens first. IO scheduler is a thread itself and subject to CPU scheduling.

It doesn't happen like the the IO scheduler algorithm is run by whichever process is queueing a request. A good way to see this is to look at crashes that have elv_add_request() in their path. For example https://bugzilla.kernel.org/show_bug.cgi?id=7667:

 [...]
 [<c027fac4>] error_code+0x74/0x7c
 [<c019ed65>] elv_next_request+0x6b/0x116
 [<e08335db>] scsi_request_fn+0x5e/0x26d [scsi_mod]
 [<c019ee6a>] elv_insert+0x5a/0x134
 [<c019efc1>] __elv_add_request+0x7d/0x82
 [<c019f0ab>] elv_add_request+0x16/0x1d
 [<e0e8d2ed>] pkt_generic_packet+0x107/0x133 [pktcdvd]
 [<e0e8d772>] pkt_get_disc_info+0x42/0x7b [pktcdvd]
 [<e0e8eae3>] pkt_open+0xbf/0xc56 [pktcdvd]
 [<c0168078>] do_open+0x7e/0x246
 [<c01683df>] blkdev_open+0x28/0x51
 [<c014a057>] __dentry_open+0xb5/0x160
 [<c014a183>] nameidata_to_filp+0x27/0x37
 [<c014a1c6>] do_filp_open+0x33/0x3b
 [<c014a211>] do_sys_open+0x43/0xc7
 [<c014a2cd>] sys_open+0x1c/0x1e
 [<c0102b82>] sysenter_past_esp+0x5f/0x85

Notice how the process enters the kernel calling open(), and this ends up involving the elevator (elv) algorithm.

Let's start with the IO scheduler first. There's a IO scheduler per block device. Its job is to schedule (order) the requests that pile up in the device queue. There are three different algorithms currently shipped in the linux kernel: deadline, noop and cfq. cfq is the default, and according to its doc:

The CFQ I/O scheduler tries to distribute bandwidth equally among all processes in the system. It should provide a fair and low latency working environment, suitable for both desktop and server systems

You can configure which scheduler governs which device via the scheduler file corresponding to your block device under /sys/ (You can issue the following command to find it: find /sys | grep queue/scheduler).

What that short description doesn't say is that cfq is the only scheduler that looks at the ioprio of a process. ioprio is a setting that you can assign to the process, and the algorithm will take that into account when choosing a request before another. ioprio can be set via the ionice utility.

Then, there's the task scheduler. Its job is to allocate the CPUs amongst the processes that are ready to run. It takes into account things like the priority, the class and the niceness of a give process, as well as how long that process has run and other heuristics.

Now, to your questions:

What is relationship between IO scheduler and CPU scheduler?

Not much, besides the name. They schedule different shared resources. The first one orders the requests going to the disks, and the second one schedules the 'requests' (you can view a process as requesting CPU time to be able to run) to the CPU.

CPU scheduling happens first. IO scheduler is a thread itself and subject to CPU scheduling.

It doesn't happen like the the IO scheduler algorithm is run by whichever process is queuing a request. A good way to see this is to look at crashes that have elv_add_request() in their path. For example:

 [...]
 [<c027fac4>] error_code+0x74/0x7c
 [<c019ed65>] elv_next_request+0x6b/0x116
 [<e08335db>] scsi_request_fn+0x5e/0x26d [scsi_mod]
 [<c019ee6a>] elv_insert+0x5a/0x134
 [<c019efc1>] __elv_add_request+0x7d/0x82
 [<c019f0ab>] elv_add_request+0x16/0x1d
 [<e0e8d2ed>] pkt_generic_packet+0x107/0x133 [pktcdvd]
 [<e0e8d772>] pkt_get_disc_info+0x42/0x7b [pktcdvd]
 [<e0e8eae3>] pkt_open+0xbf/0xc56 [pktcdvd]
 [<c0168078>] do_open+0x7e/0x246
 [<c01683df>] blkdev_open+0x28/0x51
 [<c014a057>] __dentry_open+0xb5/0x160
 [<c014a183>] nameidata_to_filp+0x27/0x37
 [<c014a1c6>] do_filp_open+0x33/0x3b
 [<c014a211>] do_sys_open+0x43/0xc7
 [<c014a2cd>] sys_open+0x1c/0x1e
 [<c0102b82>] sysenter_past_esp+0x5f/0x85

Notice how the process enters the kernel calling open(), and this ends up involving the elevator (elv) algorithm.