If you want a systematic list of tricks and what groundwork you need for each I suggest you buy a book such as Kyra Sundance's 101 dog tricks, Although she does not use a small dog as a demo nor does she use an old dog, a short legged dog, a long backed dog, and arthritic dog, nor a dog with liver problems, each dog has it's own limitations and most of them are not to do with size. As you are very focussed on the small dog aspect you might find it instructive to look at the
youtube channel of New Bear, a currently dormant member of this forum.
Okay now let's focus on the frog trick. I was not intending to say that the trick can only be trained by backing on to a cushion, I am sorry if I gave that impression, I was trying to point out that if your dog is not advanced enough to understand the back up cue then it probably will struggle to understand that it can do a trick with it's hind legs. Cricket is a dog that is very used to being trained, in fact
I rather think I recognise Cricket as belonging to a dormant member of this forum, a really gorgeous dog that has probably been free shaping for at least 3 years. The other video you showed is a dog that is also very clearly used to being trained and as you rightly point out is has a lot of groundwork already laid down, I also suspect that it naturally takes up the frog position so what the video shows is capturing rather than training the trick. If you want to train this trick using a pillow targeted by the front paws that is fine, you have decided the method you intend to use so go ahead, work it out, there will be a way and you will learn a lot through the process.
If you want advice on how
you could train the frog trick then based on the fact neither you nor the dog are experienced I suggest training the dog to target your hand/an object. Here is video of how to train paw targeting, it makes NO difference what size or age the dog is for this trick.
after the dog is comfortable with paw targeting then with the dog in a down position on a soft surface offer it the paw target, click and treat, then repeat but moving the target slightly further back each time. When the dog is comfortably extending it's leg each time slip the target just out of reach as the dog tries to touch it, click and treat. Repeat until the dog knows it is the movement of the leg not the touching of the target that earns it's reward.