You can very well show this as an experience. Don't hide your professional experience, even-though you worked from home.
If you are happy with the money you are earning for 3 days a week and are able to spend time with your family and friends, you may not be required to quit from this job. You may ask for any hike or any other benefits you required from here itself. If they refuse it, you may plan for quitting.
New company you are going to join may not provide the facilities you are enjoying here. They may make you to work until you complete the task which results in extending the shift timing, you may need to adjust with new colleagues, team leaders, managers etc.., If you enjoy taking risks, you can search and find a new job and quit from here.
All the best.
15th August 2016 From India, Chennai
The break in your career might have triggered both your present employer to offer the job on part time basis and you to accept that. Since your performance was found to be satisfactory, they subsequently made you a full time employee. Though they raised your salary commensurate with the full time employment, for obvious reasons best known to themselves only, they have not issued any formal offer letter. It has become the order of the day that 90% of the Private Educational Institutions follow such unethical practices in employment matters. So, slowly and unknowingly, sense of insecuity of tenure started occupying your mind and resulted in the questions found at the end of your post, if I were not wrong.
1) Only the wearer knows where the shoe pinches. So, if you want to quit, you can do it but before that ensure you get a good offer from a scrupulous employer.
2) I am not able to understand what you mean exactly by the second query. If it is probably your initial part time engagement and the subsequent defacto regular employment without a specific appointment order, just you have got ample evidence in the form of the documents you mentioned in the third query. If you do not hide the facts or try to twist them just for the sake of proving your continuous experience, no prudent employer would reject your candidature on this score when you are otherwise employable.
The foregoing answers will also apply to your last two queries.
If you get a better offer elsewhere, your present employer may try to lure you with offers of added benefits or threat in order to retain you. In any case, try to ensure that your exit is smooth and peaceful.
30th August 2016 From India, Salem