If the gas tank outlet is mounted above the line of the fuel inlet on the carburetor, you probably don't have a fuel pump. If the fuel tank is under the seat and rear fender area, then you will have a fuel pump, to lift the fuel to flow into the carb.

I recently pulled a push mower out of the garage. It had been sitting 2yrs. I removed the drain plug from the carb and drained it. The gas tank was empty. I filled it with fresh gas and primed the carb. It would only run 30 seconds or so. After a few failed trys, I pulled the carb and removed the bowel. There were no foreign materials or bad gas. However, I found the float pin was actually stuck in place due to a gum coating that formed from dried gas. Point is, you might need to pull the bowel from the carb and check the function of the float and float pin.

Fixed..plug in bypass port, didnt know about that, removed it and all is well. Finally heat in the shop and I learned something new.

I'm leaning towards just heat pump with heat strips. That should be fine for your area. If you plan on keeping it that warm in the house as you say in your OP, I would get the Bosch Inverter heat pump as it produces heat clear down to about 5-6 degrees ambient and very efficient at what it does. Then use the strip heat as you need it. I don't know that you'll find a contractor though for this for a new construction job as Bosch just recently came to the Houston market a few months ago. The distributor network for Bosch is in it's infancy... so there are pros and cons for choosing this brand. For what it does / how it does it --- it's a game changer for my market. (Katy, Texas area) As a comparison --- an average run of the mill single speed heat pump will probably struggle once temps drop into the 20's for any significant amount of time. They are efficient when compared to strip heat (utility bill cost wise) but there are higher limitations with those. I've gotten email notifications from my electric provider that they are estimating my next electric bill at less than $50 for November. This may wind up being my lowest electric bill ever for my power rate of 12 cents per KWH. With that said, I keep it quite chilly during the winter and my house is all electric (no gas at all). I shut it off at night and really just use it to take the chill out of the house in the morning. If it gets really cold here, I will probably just heat the main part of the house over night --- as I have it attached to a 4 zone duct system. (This is to explain the how and why my electric bill is so low -- design of the HVAC system is a big part of it.)