FWIW: I think @JohnP's answer is likely the best advice you'll get. Start with a new and proven program and go from there. His recommendations are also spot on.
This is my specific opinion on your current regime:
Tuesday: chest & calves & lateral delts
Wednesday: Rest or abs(?)
Thursday: back & calves & back delts
Friday: delts & trapezes
Saturday: arms & calves
Sunday: Rest or abs(?)
Is this a bad program? No, but it has multiple flaws. I say that because it is incredibly similar to the layout I'm using for my own training right now, however, I'm seeing a continual increase in strength. I've been doing it for a year (read: 52 weeks) and although the progress has slowed, there is still progress.
- Why are you working out calves three times a week?
I would add one, maybe two, calve workouts onto your leg day and that's it. Calves are a small muscle that will contribute negligibly to your strength increase.
- You're working your deltoids in three separate workouts!
I don't know your specifics but this also seems unnecessary. Work the deltoids on your one day and move on (same as calves).
I like that you've got rest days on Wednesday and Sunday but you mention only abs. I do hope you realize that core strength plays a phenomenal role in many workouts. My suggestion is to put two days of rest between upper body workouts and two days of rest between lower body workouts. In other words, if you're squatting Monday, don't deadlift and earlier than Thursday (you can do upper body workouts between this).
The delt workout after delt workout after delt workout is just going to get your muscles exhausted. Any delt workout is going to start targetting chest and arms. Then you work arms, and you're back to chest.
Do abs on your arm day, put some good rest between it and any other compound lift.
- Are you putting in the work outside of the gym?
The food you eat, the quality of sleep you get, the stretching/warmups you do, and the lifestyle choices you make (smoking, drinking, career) will all play a role. In a previous post of yours mentioned a high-protein diet, but are you eating enough calories? I don't know your specifics but you probably want to be in the 3000 calorie range: day after day, not just for one day here or there. If you work a demanding job, eat even more.
I purposely built de-loads into my workout. It is impossible to challenge your 1RM each week, every week. I run 12-week progressive overload where week 1 is a lightweight, high rep workout and week 12 is a low rep, high weight workout. After Week 12 I de-load and start back at week 1's light weights (although heavier than the previous Week 1).
So, my answer to
Unable to maintain weights and increase in reps from work out session to session, why? is a combination of poor programming, likely lack of rest + overworked muscles, and potentially diet.