People have been using heat lamps to raise chicks for over a century. Accidents can happen with whatever method you use to provide heat but as long as you set it up properly they work well. Heat lamps can cause burns and fires, but so can other things.
I consider heating plates and heating pads to be excellent ways to provide heat to chicks in a brooder if they are set up and used safely. I know this is an old link, I think the number of deaths from heating pads has dropped to about 5 per year lately, not the 8 they mention in this link. If I remember right the number of burns from heating pads treated at emergency rooms has grown but I did not search long enough to get current statistics. I’m not including this link to scare people away from heating pads, but I am warning people that like everything else they need to be used safely. A false sense of security is dangerous.
http://thermo-pad.com/esafety.htm
My main suggestion for using a heat lamp safely is to throw away that clamp and securely fasten the lamp with wire or chain so it cannot fall. Do not use plastic or fiber that can melt or burn, use metal and make sure it is very secure. That clamp is dangerous. And keep it away from stuff that is flammable. Give it some distance.
My ideal brooder has one spot that is warm enough in the coolest conditions and a spot cool enough in the warmest of conditions. I brood outside using heat lamps and can have huge temperature swings, going from below freezing one day to above 70 F a couple of days later. In your house with climate control it should be a lot easier to manage. I find that chicks straight from the incubator or post office are excellent at managing that themselves as long as they have the option.
That 250 watt bulb is more than you need indoors. One risk in a heat lamp is that you overheat the chicks in a relatively small brooder. I use 75 or 125 watt bulbs in my outdoor brooder in summer but those can be hard to find. I had to order my bulbs online. I like the red heat lamp bulbs and all the feed stores or hardware stores carry here are the 250.
Size of the brooder is also important. At six weeks those chicks will be a lot bigger than they are now. I don’t know how big your brooder is now but you might want to think about some options to increase size if you need it. Some people will tell you that you need more room in your brooder than you need for your coop but I don’t find that necessary. For 15 pullets I’d suggest a minimum size of about 12 square feet, maybe a 3’ x 4’. Bigger is better. If you keep them in there much past six weeks that could get tight pretty quickly.
It’s generally a good idea to keep them locked in the coop only for about a week after you move then to the coop before you let them have access to the run. That gives them an idea of where home is. When I open the pop door to give them a chance to go into the run, sometimes all of mine are on the ground within 15 minutes, but usually not. Sometime it may take three days before they all build up the courage to go outside. I don’t rush mine, I let them make up their own minds. It can be hilarious to watch them decide if it is safe or not. Many things about chicken TV can be better than anything on cable, satellite, or antenna.
Sometimes they don’t automatically go back in the coop to sleep when it gets dark, mine often gather next to the door even if I’ve kept them in the coop for over a week before I let them out. I wait until they settle down for the night and lock them inside. Sometimes they get the message the first time, I’ve had some that take over three weeks of tossing them inside. Usually mine all get the message after about a week. You’ll find that each brood is different, you can never tell for sure what any of them will do.
Good luck and welcome to the adventure.