No, definitely no. :D

When people say that something happens in Russian to make pronounciation easier, it's usually because the real reason was forgotten. ^^'

Есть is the only form of the verb быть used at present, but it is more used with existence sense.

У меня есть is something like "at me there is" = I have. Есть яблоко на столе = there is an apple on the table.

This form is also used like this: Бог был, есть и будет вовек. = God was, is and will be forever. When you want to say that someone simply is, since you don't use быть in the present tense, example: Он мужчина = he is a man.

I hope this helps you.

• У меня есть...

There are two different "есть" :) One you've mentioned and another one means "to eat" (infinitive):

I want to eat. - Я хочу есть.

In a general declarative sentence, new information usually comes after known information. I think it is the same as in English. Example: This is a tree. The tree has leaves. The leaves are green. Green is a color.

You can stress important information with intonation regardless of word order:
Он здесь ест. He is the one who eats here.
Он здесь ест. He eats here, not there.
Он здесь ест. He eats here, he sleeps anywhere else.

There is an old Russian joke (or rather one of its many variations):
Stirlitz has a private word with a German officer: "Herr Müller, would you like to work for the Soviet intelligence? We pay well."
Müller: "What are you talking about?! Of course not!"
Stirlitz: "Oh, in that case, do you happen to have something against a headache?"
Stirlitz was smart enough to know that people tend to remember only the last sentence of a conversation.

So you can change the word order to put the accent on something. It is not a rule, but you usually put the important things closer to the end:
Он здесь ест. He eats here. This place is not intended for eating.
Он ест здесь. He eats here. He sleeps over there.
Здесь ест он. He is the one who eats here. Go find your own place to eat.

However, you may make yourself sound like Yoda if you mess with the word order without caution, trying to put the important part in the end in every case.

Pronouns are being left out if the meaning is clear from the context or the verb ending. For example, "ем" (am eating), "иду" (am going), "сделаю" (will do), etc. can be understood only as "я ем", "я иду", etc., because of the verb form. The same goes with "ты" (ешь), "мы" (едим), or "они" (едят). On the other hand "ест", "идёт", "сделает" have the same form for "он" (he), "она" (she), "оно" (it), so you cannot just drop it without adding some ambiguity.

Finally, in general, you leave the pronoun out only in simple sentences, i.e. Иду. I am coming.
Думаешь? Do you think so?

Though, you may want to omit "я" or "мне" when you add some politeness to your speaking. There is a saying "я — последняя буква в алфавите", meaning that it is not good to be egocentric and use "я" often in your speech (at least colloquially). I.e. "(я) думаю, это не так" (I think, it is wrong) or "(мне) кажется, нам направо" (I think, we should turn right).

On the other hand, mentioning "я" makes your words more weighty. So it is usually present in the statements like "Правильно, я считаю" (It is right, I think (sure)).

Oops, есть was an swipe keyboard typo.