Hi, I reccommend laminate flooring as it's so much easier to deal with & costs the same as carpet these days. My husband used a small jigsaw to cut intricate shapes around the door frames. You just need to measure & pencil onto the wood making sure you leave room for mistakes, or just start on a new piece of wood! This takes time & patience (which my husband has none of either) & after using a few pieces for the first door frame he got the hang of it. It doesn't need to be absolutely perfect as you can buy a range of wood-coloured flexible sealants that you sqeeze into the gaps. The sealant allows room for shrinking & swelling of the flooring as the seasons change & prevents any debris filling the gaps. You can then attach your skirting boards to cover the gaps (or attach the laminate strip to your existing skirting board which is what we did).
I slide the laminate and underlay up to the architrave and mark along with a pencil, then I cut the architrave so that the laminate slides under it, there's no cuts and no gaps, leaves a well finished job.
The best tool to use is a Fein Multi master tool. It has several different blade attachments, and cuts very neat with no gaps at all. We wouldn't recommend using a fillers as these can harden and all laminates need to be able to flex or they will get warped damaged.