Using a plastic dog create

[Guisso14]

At this point, I don't think you need a heat lamp, especially not in the coop. Actually, I think what other posters mean by light is "daylight." I can see that you're trying your best and doing a good job, but I can offer some suggestions that may make life easier...

Newspapers can be slippery for chicks and very hard to keep clean. I would say get rid of the paper and use pine shavings or some bedding material instead. Raise the food and water up higher, like on a brick or something to that effect. It'll stay cleaner and spill less.

You probably won't want them in the house much longer, they can go out to the coop right away, even if it's cold, as long as you make a few provisions... which will require (electricity) running an extension cord to the coop. I made a heating pad cave for my babies and many other people have also successfully and easily raised their chicks this way. It's similar to an expensive "heating plate" and works like a mother hen. The chicks go underneath for warmth when they feel like it, and run around and explore the rest of the time. They're exposed to natural day and night cycles and feather out quickly.
Please read about it all here:
Mama Heating Pad
Kick The Heat Lamp: Better, Safer And Healthier Options To Heat Your Brooder
Raise chicks outside

You could easily use a few layers of that poultry wire and shape it into an arch, securely attach a heating pad (make sure it will never automatically shut off), then you can adjust the height as the chicks grow by just bending the wire. Here's some pics of my set-up to get an idea (but I used a cooling rack with adjustable supports):
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I actually moved them out to the coop right after those pictures were taken (2 wks old).
And I put cardboard on the top of the heating pad to protect it from poop.
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Maybe this can help solve your space problem, by just moving them to the coop now.
Good luck and have fun raising your flock!!

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My plan for this weekend is to set up the coop with a heat lamp, I have a built on run for them and just letting them in and out during the day and closing it up at night. I have a cousin who owns a wood mill so I can get an assortment of wood shavings from him free. I'm just using paper for now because it was easier to clean up. It doesnt look like Much but It has tons of space inside the coop (for 4 chickens) and the run is about 4x8 with space under the coop.

I'm gonna try moving them out with a heat lamp this weekend.