Ok, so in order to resond to your suggestion I'm gonna need to break it up.
The first issue is the energy prices for exploring. Your pets are assigned ten energy a day and five of it is wasted on exploring and trying to level up your pet. I would like there to be maybe more energy.
Ok, understandable you want more energy. However, changes were just recently made because the game was too grind-happy. So, yes you use more energy, but you gain just as much exp and stuff. Since you're a new user you don't have that many pets so it's easy to go through all of their energy. However, the free maximum for user pets is 200. Once you have 200 pets I guarantee you wont be complaining that you don't have enough energy. Also, it's not 10 energy per day, it's 1 energy per hour so technically a maximum of 24 energy per day. If you only get on once a day then yes, you only get the benefit of 10 energy per day but you could be getting up to 24.
another issue is the time it takes for catching pets and doing quests. Four to three hours feels like a long time to waitfor catching and quests even half an hour would be a nice enprovement to the wait.
Keep in mind this isn't a game you beat. People are expected to play this game over a period of months and years. I assure you that 4 hours for a quest is nothing really since you can essentailly set-and-forget it. You probably don't have too much money yet but once you start amassing coin and get further into the game with more pet breeds and genes you'll easily be making more money than you need to play the game. Taming and breeding take a long time because, as I said, the game is meant to be played over months and years. If you could just breed, bam have a baby in 30 minutes, you could easily make perfect pets in a few days. Then what would there be left to do?
The last issue is energy regain items while yeah all of the accounts are premium right not most average users would not be able to afford the energy regain items currently. So Maybe cheaper energy regain items would be nice and a wider diversity of them.
This is sort of the same issue as #1. Energy regain items are not meant to be used willy-nilly or regularly. Once you have more pets you'll be able to explore ALL you want. (Personally I can explore for ~4 hours straight with how many pets I have but that just wears me out!). As for battling pets you really should focus on no more than 10 at once (if you intend to get on the highscores board at least) because you can't get rewarded for more than 10 a week -- and that'd be taking the ENTIRE highscores board top 10 (very unlikely!). As a point of reference, I took 6 spots of it last week and I was working my butt off to ensure that my energy was never capped and that I was winning as many battles as I possibly could (which means needing to find enemies 5 levels below you very often!)
So, pretty much all your concerns will be fixed by getting more pets, levels, and better genes. Just remember this game is fairly casual with no definite end or absolute goal to win, so take your time and have fun! Stop worrying too much about being the best of building all the buildings RIGHT NOW. They'll still be there when you finally get around to making them; the rest of the game isn't going anywhere either.