いきなり is an adverb meaning "out of nowhere", "all of the sudden". Forget "without warning" for now. It plainly modifies 言い出す as an adverb. 言い出す is the first verb after いきなり, so it cannot be simpler.
何を is "what". 何 is the object of 言い出す.
言い出す is "to start saying", "to bring up (a topic)". Its subject is the girl.
か is the question marker.
と思えば is the plain ...
I think the part you're missing is that なくし is the 連用形 of 無くす. The 降られ works like this:
降られる = 降る + passive
降られ = 降る + passive + 連用形
So both 降られ and なくし are 連用形 used to connect clauses. This is how you should parse it (a bit on the liberal side for the translation):
I'm rained upon and I have no place to ...
I think this is in fragments... 雨に降られ is like having (or with).. the rain fall(ing) on me.. (passive of falling - here like it happened to me) 行き場なくし, losing (giving up) my place to go, なんの罰さ what did I do wrong.. (what mistake did I make)... couldn't say without more context.. but the lyric seems to be more of the speaker's train of thought.
In this particular case, both この自然が多い町 and 自然が多いこの町 refer to the same thing, and they are interchangeable.
In many other cases, however, placing この at a distant place may introduce a difference in meaning:
This town where fish are delicious
The town where this (particular) fish is delicious
this town where I met my wife
I can't understand why and how come both modifiers are put together directly in this fashion: 自然が多いこの町...
In the phrase 「自然が多いこの町」, 自然が多い is a relative clause that modifies この町.
This town ［which has a lot of nature /which is rich in nature］
< Its non-relative version is:
この町は、自然が多い。 This town is rich in nature.
Is it possible to join ...
“I just woke up and I work later.”
今起きたところなので、もう少ししたら仕事に出かけます。(If later means few minutes. Sometimes 「もう少し」means more longer time, like one hour later or so, but it depends on 文脈 or situations.)
しばらくしたら仕事に出かけます。(same as or a little longer than 「もう少し」)
今起きた所です。何時間かしたら仕事に出かけます。(if later means few hours)