One of the things that I do for a living is that I am a house painter and decorator.
There are a number of possibilities as to why a paint job can be peeling - poor surface preparation such as not cleaning the surface or removing loose paint, not using a primer first, quality of the paint used, applying a paint over that to which it is incompatible, etc.
Please check out the lead thing first: Removing Paint from Exterior Surfaces to Prevent Exposure to Lead: Environmental Health - Minnesota Dept. Of Health
Surface preparation and the kind of paint that you use are paramount to a successful paint job.
After checking out how to deal with the lead paint, I would wear a good dust mask and scrape the peeling paint off.
It sounds like you want the job to look as perfect as possible because you are concerned about it looking patchy where the edges of the old paint show through. They will show through if you don't smooth them out entirely. I would not advise you to use a heat gun because of the danger of fire - I have seen this happen. But, I would suggest using a palm sander to smooth out all of the edges. You can find palm sanders that are not expensive and that will last you a long time. Got my current one at a Farm & Fleet store for about $20.00. Cut the retangles out from larger pieces of abrasive (sand) paper. I prefer to use a palm sander rather than one that goes in a circle because you can see the circle marks underneath the paint. Also, with another type of sander such as an inline one, there is the possibility of removing too much of the material, such as your wood, too easily.
I prefer to brush the paint on because it goes on thicker than if you spray it on. Spraying puts it on too thin. I also like the "warm" look to the paint job that brushing it on leaves. Must make the paint strokes as straight as possible.
Since you have wood siding, I would use a 3 inch high quality paint brush that tapers out to a feather edge for the largest surfaces. Then a 2 inch and 1 inch brushes of the same quality that have bristles set at an angle for the areas of trim.
For durability, I would use at least a satin or semi-gloss paints for the large areas and a semi-gloss for the trim areas. They will last longer than flat finish paints. Consult with the paint store about this. It may not be desirable for the large areas of your home to have any sort of glossy look.
Use a primer first, and apply at least two, thin coats of the paint. There really is no such thing as a one-coat paint - it is just a way to sell the paints. Unless it is applied over exactly the same color and type of paint the was previously used - recently.
For my customers and stated in my contract, I must use either Dutch Boy or Sherwin-Williams paints in order to guarantee my work. Sherwin-Williams owns Dutch Boy. About six years ago, I used a Dutch Boy semi-gloss latex on outside portions of my home. They still look like they have been freshly painted. There are a couple of other high quality brands of paints, but the ones that I have mentioned are my preference.
Keeping track of where you have painted can be tough. A gentle overlap of paints is okay.
I hope that there was a contract with the previous painter that spelled out some kind of warranty, I.e. At least a year to be free from...
Anyway, without going into a lot of other detail, here is a link for some more advise. Please let me know if you have any more questions. Thank you! And, I wish you well with your paint job! How to Paint the Exterior of a House - eHow.com