I have been trying to explain speak more simply and naturally in everyday English with a by using shorter sentences. As Albert Einstein had has been quoted as saying: that "if you can't explain it simply, you don't understand it well enough." [This should be two sentences. As you suggest, simple is better. You used "had" intead of "has" — it is present tense because you are quoting what he said in the past; the action of quoting is happening now].

Here is my attempt of at some different ways of describing the same situation in two different ways in English:  to improve my English skill.

- I just want everyone to know how grateful I am for your well-wishes and helpful advice. All of your words cheer me up whenever I'm feeling down. I really appreciate your time for in generously sharing all of your common experiences. [Techincally, "well-wishes" is not proper English. However, it is commonly used. To be precise, you should say "good wishes" (since good is an adjective, unlike well, which is an adverb) but people might think it sounds less natural]

-Thank you for taking the time for we shared to share all of your common experiences and express your well-wishes [or "words of inspiration/kindness" or "kind/inspiring words"\. It really means a lot to me, especially when I have been feeling emotionally drained.

-I received a cookbook for dogs and a handmade mat as for my dogs gift as well.
-I also received some gifts such as a cookbook and a handmade versatile mat for my dogs. [This sentence is much better].

-This mat will certainly be coming everywhere with my dogs and I  to everywhere.
-This Leopard-print mat will certainly be coming everywhere with my boys and I/them to everywhere.
-This leopard mat will definitely be one of my dogs'/their must-have items to pack for when going on a trip.
-This leopard mat will fast become my boys' their favorite going-out stuff item. [or "favourite item for when we leave the house" or "favourite item for journeys/trips away from home|
(Note; What I'm trying to say in above is that I will put that handmade gift from my friend into my dog's stroller or travel carrier as a cushion and they will love it.)

Thanks for reading and your help is very much appreciated.
Thank you for taking your time to read my practice notes.

Remember, as Einstein said, to keep your sentences short and simple! Your shorter sentences are easier to understand and it's easier for you to convey what you're saying. Well done Ai! Your dogs are very cute.