Struggling with Dativ and Akkusativ and Der-Die-Das-Den-Dem

I attended German class this morning as usual for Tuesdays and Wednesdays, and was honest with the teacher: "It's a total guess for me as to which article to use!"

In German it's not just
"the" as in English.
The dog, the door, the bed, the tables. Easy, right?

And it's not as simple as French, with only
"le" (masculine: le chien) or
"la" (feminine: la porte) or
"les" (plural: les tables).

No, no, in German, it's
"der" (masculine in nominativ case: der Hund; or feminine in akkusativ case: der Tür), or
"die" (feminine: die Tür; or plural, die Tischen) or
"das" (neuter: das Bett), or
"den" (masculine in Akkusativ case, or plural in Dativ), or
"dem" (masculine or neuter in Dativ case)

And of course half the time what's masculine in French is feminine or neuter in German and vice versa:

e.g. the table: la table (fem in French), vs. der Tisch (masc in German)
or the bed: le lit (masc in French), vs. das Bett (neuter in German)
or the cat: le chat (masc in French), vs. die Katze (fem in German)

Help, help! I can usually figure these things out if I have time and solitude:
1. I look the word up in the dictionary (whether paper or online) to find out if it's masculine, feminine or neuter
2. I think about whether the thing is the subject (nominativ) or direct object (akkusativ) or perhaps an object of a preposition (akkusativ or dativ)
3. I think about whether the action is taking place in one spot (dativ case), or whether it involves moving from one place to another (akkusativ)
4. I look up in the tables (whether on paper or in my head) which article is thus the correct one to use

It's quite the multi-step process just to pick one little tiny inconsequential word before a noun. But on the spot, in class? I blush and sputter, or just pick randomly, and the teacher corrects me - but what use is that?

Ah well, I know I am learning a little each class, and with each piece of homework. David shared a great little trick with me today which helped oodles. "Wo?" (where) and "Dativ" are the shorter words and go together, and "Wohin?" (where to) and "Akkusativ" are the longer words and go together. Thanks, my love! Helps no end.