Some languages don't need suffixes like "at" or "in", or "the" especially to describe times of days, while some languages are heavily dependent on it (e.g. la manana vs manana in spanish, am Morgen vs morgen in German).
For example, in some languages: I come morning, is the equivalent of saying I come in the morning.
No, it wouldn't be. But you can use 'sabahleyin'. In fact the corresponding expression for 'in the morning' is 'sabahleyin'. 'Sabahta' is not used at all. That can be counted as irregular. There a few words like this: akşamleyin(in the evening), geceleyin(at night), öğlenleyin(at noon), yazın(in the summer), kışın(in the winter)...
These are what i remember.
That sounds like folk etymology. It's a good way for a native speaker to rationalise the random homophony and a good mnemonic for us students, too, but I don't think they are actually related.
Now, it's surprisingly hard to find reliable and comprehensive sources on Turkish etymology (especially when your Turkish is not good enough to do the search in Turkish) but the English Wiktionary seems to indicate that they are not related (the only difference in the parent words seems to be vowel length, still, they appear to be discretely distinct) and this etimolojik sözlük appears to agree.
If you're talking about that you usually/always open the restaurant in the morning you can use both 'açarız' and 'açıyoruz'.
If you're talking about that you're going to open the restaurant for example tomorrow and planning, saying 'Let's do it like that.' you use 'açarız'.
f you're talking about that you're going to open the restaurant for example tomorrow and it is planned already and exact what will be done you use 'açıyoruz' or 'açacağız'.
If you have any other scenario in your mind, i'd like to answer.