Highly unlikely. They are separate systems and can't really influence each other.
The most likely problem with the heater blower on your 2003 Sable is the blower resister. The resister can go bad and cause you to only have one speed.
If "heater coil" is just another term for "Heater core", then yes, most all cars have them.See "Related Questions" below for tons of information about Taurus / Sable heater cores.
Right side
Do you mean that your heater core is leaking coolant ? That would mean that the heater core is clogged. The heater core is like a mini-rad, if it is clogged it leaks.AnswerNo, there is not an overflow tube/outlet on the heater core.See "Related Questions" below for more about Taurus / Sable heating/cooling systems.
remove the entire dash
The "Related Questions" below provide a comprehensive trouble-shooting guide to Taurus/Sable heating problems - particularly the inability to adjust the temperature and flushing a heater core.
Look behind the glove box. under dash.
Heater core.
No one knows if:The heater is NOT blowing - or blowing on high speed only, or... The heater IS blowing, but the air is NOT hotEither way, the "Related Questions" below will get you started
The heatercore itself is more centered under the dash.The blower motor is behind the glovebox.See "Related Questions" below for an excellent write-up on the 1986-1995 Taurus / Sable heater core
The heater core could be clogged up.
A bad heater blower motor resistor would effect one or more blower speeds other than high.