You can use 'will' as you did in your example, but it changes the meaning slightly. It's still about the future, but you are adding something extra. With 'will', you are more saying, "I intend starting my new job on Monday.' This isn't quite the same as saying, 'I start my new job on Monday', as you're adding an element of intent.

I've never heard about any special rule for those verbs. In general, we CAN use simple present tense when we're talking about an action in the near future, or an action that will happen at a very definite, specific time. Future tense is also okay, but simple present is common. For example: 1) "I fly to Paris tomorrow." 2) "My son starts school next month." 3) "She turns thirty this year." We use the future tense when the action happens in the more distant future, or is less certain. 4) "I'll fly to Paris after the holidays are over." 5) "My son will be starting school two years from now." 6) "If you're twenty years old today, that means you'll turn thirty in the year 2029." 7) "When will you finish writing your book?" There are some exceptions, though. You can see some good examples and links for more information in Chris's comment to this question: https://www.italki.com/question/479185

The others have said it - there's nothing wrong with 'will start' and 'will finish.' But you need a reason to add in the 'will' - usually it's because you want to add emphasis. I WILL start on Monday. Another reason is if it's a part of a spontaneous thought or dialogue. "I think I'll start on Monday. Yes! Monday! What a good idea!" Generally don't put 'will' in unless you have a reason to.