we went throuh the period ___ communications were vary diffi

Slanting through the trees,yesterday's late afternoon sun (cast) more deeper shadows on the bedroom wall.
　　前面的主语是 yesterday's late afternoon sun,虽然句中没有明显表示过去的时间状语,但是“昨天下晚的太阳”暗含了太阳是在过去投下了阴影,因此应该用过去式 cast（cast 是不规则变化的过去式还是cast,casted 是错误形式）.
　　正如你说的,此题的的错误出在双重比较 more deeper上.

2.Recent studies indicate that eating chocolate and other sweets neither causes nor aggravates ache.

　　上面的句子和 Recent studies indicate that eating chocolate and other sweets does not cause or aggravate ache 在语法上都没有错误,但是表述的意思不同.
neither causes nor aggravates ache 是全部否定,意思是“既不引起疼痛也不加重疼痛”,二者都不存在.
does not cause or aggravate ache 是部分否定,意思是“不会引起疼痛或者不会加重疼痛”,二者必居其一.

1. 爱的最高形态可以被达到（触摸到）,一旦你们度过那些平淡的日子.
reach 够到, once 连词, 一旦
2. to follow 不定式做定语 修饰 the hardest thing
比如 The first person to come is Mr Li. 第一个来的是李先生.