"你多吃一点" literally means "eat more plz" and "慢慢吃" means "eat slowly" , which are commonly used when a host asks guests to help themselves. They are the cliches ppl use to show hospitality, which have lost much of its original meanings through overuse. But on second thought, i think they might imply that there's more than enough food to go around, so plz eat more n eat slowly:-)

It's Chinese custom when I finished my meal and you haven't, I would say "慢慢吃" which means "Enjoy your meal".

these are something like "take ur time" "no need to rush" "enjoy ur meal" in english, a way to show their hospitality, u dont really need to eat a big meal if u dont wanna get fat ;)

It is a way to show their caring of you and it also use out of polite or friendship it is something like English love to ask about the weather

These are standard phrases of social politeness and are not supposed to be taken literally. The sentiments that are intended are: 你多吃一点: We are so happy to be treating you to lunch/dinner. If a mother says it to a son, it means "I care about you." 慢慢吃: Please enjoy.