Help with a REAL keto diet...

They are "allowed", although some people feel they increase appetite.  You may get a better physical response by switching as much as possible to natural non caloric sweeteners like Erythritol (monkfruit), Xylotol, and stevia.  Some people argue against even those, on the basis that it keeps your sweet tooth active.  Personally I don't see any problem with that.  Although we are all on the same road, we travel it at our own speed and in our own way.  Make the changes you can.  Don't corner yourself into a plan you can't enjoy.

In answer to your later questions, I have a gym membership but rarely go.  Most of my exercise is working around the house, I'm cleaning out closets and repainting the living room right now.  I sometimes take my dogs for walks but it's such a huge production that I don't do it as often as I should.  My daily food routine is 2 cups hot tea, usually with HWC and Erythritol, until around 1pm.  About half the time I have lunch at 2-3 pm, usually some salami and cheese or a spinach, avocado and egg salad with blue cheese-MCT-ACV dressing.  I skip lunch half the time.  At 6-7 pm I have dinner, 4-6oz meat (steak, chicken breast, pulled pork, salmon) with some kind of sauce, 4-6oz vegetables (cauliflower, broccoli, zucchini, cabbage, mushrooms) with butter or a sauce.  After dinner if I need to balance my macros I have a couple of fat bombs or a cup of chicken bone broth with lemon juice, collagen, and seasonings.  I've been on keto since the beginning of October and have lost 43 lbs.