I take it you have no experience with oscillating tools then. 
Wood chisels are for wood; cold chisels are for metal.
there are a lot of suggestions on tools to buy and methods to use. most all the suggestions will work. they all have been used depending on the tools a person has at hand at that particular date and time (some tools have only existed affordably in recent times).
for a person that has no power tools and few hand tools, then likely little tool experience, some of these suggestions aren’t the best. metal cutting/grinding with power tools requires safety procedures which an inexperienced person may not be familiar. a person with no power tool experience will not have experienced controlling torque of a tool. even using a hacksaw to cut a slot flush requires a short blade and holder and practice.
both the safety of the user and the quality of the job are issues. some people might not be suitable to do some things, especially if it is a one time event…
You sound like a chick.
Nothing wrong with hiring the guy to do it right. I fix stuff every day messed up by some handy man, Fireman, or DIY disaster.
Door replacements are one of those things that can go smooth and quickly but usually involve little surprises, annoyances and struggles that can take all day.
I actually have a lot of tools and a pretty good idea of how to use them. I even rewired our chicken house once, but I love my family too much to allow them to live in a house where I have worked on the electrical system.
I work on mine all the time, but then I was trained as an electrician way back in the day. My latest triumph was fixing a floor fan switch, which saved us about $150 for a new fan.
Might I suggest a square-shanked screwdriver (properly sized to the new slot) and a crescent wrench? In case the lag screw is stubborn, this will give you the leverage needed to break the screw loose.
With the new ossilating tool you can cut out the mortar in the surrounding bricks and attack the screws with everything on a workbench. Then learn a new skill of repointing the bricks when you replace them…
Seriously, I have the tools and experience to do 90% of the suggestions here. But I wouldn’t try to replace the engine in my car. Is it worth a few hours of aggravation or just call a pro?
I would have to be seriously aggravated before I’d pay someone the going hourly rate to remove a few screws. Assuming you could even find someone to come to your house for such a small job.
Presumably, he’d be hiring someone to take out the old door and put in the new one. I’d think you could find someone to do that.
Besides the rust do you have a specific reason to replace the existing security door? Rust can be removed and paint is cost effective. If the old door still meets your functional requirements it can be made to look just fine far cheaper and easier than replacing it.
Just a thought.
I was a little vague. From the point of view that someone with limited tools and experience might as well have someone do the whole job rather than was an entire day screwing around.
Then again I’ve spent a few hours making a bolt like object on a lathe rather than going to the hardware store and spending less than a buck. The journey and satisfaction of doing the stubborn way has it moments.
And band-aids, don’t forget the band-aids.
ow.
So true. I often chastise myself for being over prepared and then tend to go in the opposite direction. But I was very happy that I over did it on the bandaids when I had to bandage my foot recently. I found I had gauze bandaids of all sizes up to several square inches and could use one that was just the right size for the wound on the bottom of my foot.
I even had surgical tape to bind it to keep on tightly and therefore clean. So yes, always remember the bandaids - one of each. 
If you can afford a new door you can afford a few bucks for a pair of vise grips. Or twenty bucks or so to buy a cheapie angle grinder.
The head doesn’t allow much torque to be applied when installing, so they are easy to get out.
You can cut (hacksaw or chisel or file , or some sort of cutter or gouging device … ) , a bit of a track in the head to allow a sharp flat blade scerwdriver to grip the head. It only has to be a tiny scratch and it provides the screwdriver grip, just as long as the screw drivers tip isn’t rounded off or too soft. (you just need a quality screw driver with hardened tip …)
A drill could make the gap that the screw driver could be inserted into.
But if you can get plies around the head, that works too. You might lever , hammer , gouge or drill the edge of the head to create a shape for pliers to grip to.