How to check if a service is running via batch file and start it, if it is not running?

To check a service's state, use sc query <SERVICE_NAME>. For if blocks in batch files, check the documentation.

The following code will check the status of the service MyServiceName and start it if it is not running (the if block will be executed if the service is not running):

for /F "tokens=3 delims=: " %%H in ('sc query "MyServiceName" ^| findstr "        STATE"') do (
  if /I "%%H" NEQ "RUNNING" (
   REM Put your code you want to execute here
   REM For example, the following line
   net start "MyServiceName"
  )
)

Explanation of what it does:

1. Queries the properties of the service.
2. Looks for the line containing the text "STATE"
3. Tokenizes that line, and pulls out the 3rd token, which is the one containing the state of the service.
4. Tests the resulting state against the string "RUNNING"

As for your second question, the argument you will want to pass to net start is the service name, not the display name.

To toggle a service use the following;

NET START "Distributed Transaction Coordinator" ||NET STOP "Distributed Transaction Coordinator"

You can use the following command to see if a service is running or not:

sc query [ServiceName] | findstr /i "STATE"

When I run it for my NOD32 Antivirus, I get:

STATE                       : 4 RUNNING

If it was stopped, I would get:

STATE                       : 1 STOPPED

You can use this in a variable to then determine whether you use NET START or not.

The service name should be the service name, not the display name.