According to the common spelling logic in Russian soft and hard sounds are spelled by the same letter, sounds become soft when their letters are followed by е, ю, я, ё, и, ь. So, we can dispose of щ completely.
Here is the acoustic way of spelling: Ши да каша пишя наша. Слышыш?
No, you did not train your ear properly yet. Vowel always makes a syllable. и is a syllable, й is not. ое is two syllables, ой is one. Even when the syllable is not pronounced clearly, it is received by the mind of native speaker. You can try to read Russian classical poetry paying special attention to meter. Meter is very important even in normal speech although it is irregular. Even the wisest phrase can not become catchy without proper rhythmic metrics in Russian.I can't tell much difference between и, й and unstressed ое and ой so I'd make them all one letter.
Yep, it can be replaced by й. Подйезд, отйявленный.I'd get rid of ъ too.