You can link your HELOC to your checking account, and just ACH (or wire, if you want) the funds back-and-forth as much as you need to. No reason to use paper checks for that.
Not mine. Other HELOCs have that feature, but not my HELOC, unless I create checking account with them. They won't allow external accounts.
If you have checks against the HELOC account, presumably there is a routing number and account number associated with it? Can you try linking to that, from your other/external checking accounts?