Do I need to extend my J1 visa for these three months if I'm not going
to travel to the US during that time?
As long as you're not physically present in the US - you can do whatever you want. You don't need US government's permission to work for someone in your home country. However, you need to check the laws in your home country on that. You'll be probably considered as self-employed, so you'll need to check local laws about taxes, national/social insurance, etc.
Is it easier for the future company that sponsors me (if any!) to
transfer a J1 visa than creating a new one (H1B I guess)?
Either way they'll have to apply for a new visa for you, J or H is up to them and their lawyers. If they're exempt from cap (which, for university position, it is usually the case) then they will probably not care so much other than following their internal policies. For you however it may be important, since if you come with a spouse - the main difference between J1 and H1 would be the EAD for the spouse (J2 are eligible, H4 are not).
If the prospective employer is not a university, then J1 is not an option for them, unless these are very specific circumstances. For most commercial employers, foreign workers can only work on H1b (or similar visas, like TN for Canadians, or E3 for Australians).