ill join

I read a book 読みます I am reading a book 読んでいます

甘さ　甘み　苦さ　苦み umm good question! I think 甘み　苦み are used when talking about taste but you can use all of them , actually i think you don't have to care about the difference

this で means 'with' but ゲームをするis better than ゲームで遊ぶ i think. ゲームを遊ぶ sounds a bit off, though I can understand what you want to say

のが is useful. But Do you have a smaller one? is もっと小さいのはありますか？. I don't really understand たい焼きを食べるのがつまらなくない たい焼きを食べることが好きです。this is a written language and this can be たい焼きを食べるのが好きです in spoken language, you can use のが instead of ことが

彼は徐々に歩けるようになってきた。空が次第に明るくなってきた。 these two words are mainly used in written language, and sound a little bit difficult. In this context, i think 少しずつ　or だんだん　is more common and often used. 彼は少しずつ歩けるようになってきた。空がだんだん明るくなってきた。also, you can use them in conversation. ぼちぼち is little by little, but it's used when someone does something, like "i will do it little by little" and this is not so common, i've never said this word. As for 漸次,、i rarely see this word.

Yes. I'll give you a hand

うちに行きて、ねみませんでしたか　this does not make sense, i can't infer at all