- ANSWER TO YOUR QUESTION 1:
You have the two sentences -
1. We started early.
2. We arrived at noon.
Now you have to join the two into a single sentence. You can do it as you said in your question in one of the two ways -
3. Starting early, we arrived at noon.
4. Having started early, we arrived at noon.
They both (sentence #2 and sentence #3) are grammatically correct, and mean in this particular context essentially the same thing. But in other contexts it is not the case.
Sentence #3 can be interpreted in many ways - temporal, causative etc. In temporal domain, it can either express co-occurrence or the occurrence one after the other. But in sentence #4, using having the speaker is explicitly saying that the event expressed by the verb (start) in subordinate clause (Gerund-Participle clause - having started early) is actually took place before the event expressed by the verb (arrive) in the matrix clause (we arrived at noon). But problem might arise in other contexts.
An excerpt from From Longman Grammar of Spoken and Written English by Biber et al. Page No. 782
10.2.1.9 Overlap and ambiguity
Although many circumstance adverbials clearly fit only one of the
seven major semantic categories, not all occurrences of circumstance
adverbials are so clear cut.
first, there are many cases in which adverbials fit primarily into one
category, but have secondary roles that fit another category. manner
adverbials in particular often include aspects of another semantic
category; for example, slowly and quickly in examples below are
not only descriptions of the manner of an action, but can also be
interpreted as describing duration:
I've started but it's going rather slowly. (CONV)
They evidently expected him to go quickly. (FICT)
Other manner adverbials can include a meaning of extent/degree:
They ahve no desire to investigate this matter properly. (NEWS)
The disease pattern has changed radically. (ACAD)
In addition, certain adverbials have extremely ambiguous meaning. The
ambiguity in the use of just as restrictive and extent/degree was
noted above. Ing-clauses often present an even greater problem for
interpretation. These clauses typically have an implicit and somewhat
ill-defined relationship with the main clause. Consider the following:
- Watching him as the days went by, the guilty collector had noticed signs of physical and moral decline. (FICT)
- Three weeks ago Swedish and Scottish police searched Talb's flat in Uppsala, removing fifteen bags of clothing. (NEWS)
- The result of the operation is placed in the accumulator, destroying its previous contents. (ACAD)
In 1, the adverbial clause could be interpreted as showing a
concurrent time relationship (i.e. while watching him, the collector
noticed the decline) or as giving a reason (i.e. because he watched
him, the collector noticed the decline). In 2 and 3, the adverbial
clause could be interpreted as describing a result, a concurrent time
relationship, or an event that happened in a time sequence. The
distribution and uses of this semantically ambiguous form, termed
supplementive clause, is discussed further in 10.2.8.1 and 10.2.8.3.
Circumstance adverbials can also serve functions similar to linking
adverbials. Much of the information in circumstance adverbials creates
cohesion with information that has come before. for example, the time
adverbials then and meanwhile show the connection between the
events in the previous clause and the subsequent clause:
He planked the bottle on the table, and shambled muttering round the corner. Then he put his head back into sight.
The 21 sambas originally submitted were whittled to one. Meanwhile, seamstresses and tailors all over Rio made costumes. (NEWS)
With adverbials such as these, the connective function is made
semantically, through the circumstantial information which indicates
time relationships. Thus, they are still categorized as circumstance
The circumstance categories of addition and contingency also
occasionally exhibit similarities with linking adverbials (see
10.2.1.6 for discussion of the former, and 10.4.1 for linking adverbials in general).
There is another one I want to quote:
This example sentence is From Practical English Usage by Michael Swan (2nd Edition) Page No. 406
Putting down my newspaper, I walked over to the window. ( = After I had put down my newspaper...)
- ANSWER TO YOUR QUESTION 2:
We starting early, arrived at noon. [WRONG]
This sentence is grammatically wrong.
starting early can be a reduced relative clause or a Gerund-Participle clause. If it is a reduced relative clause, it has a progressive meaning - We (who is) starting early arrived at noon. And this is not the intended meaning here.
If it is a Gerund-Participle clause, the subject of the clause is - we. That is fine. This clause along with its own subject doesn't go with the rest of the sentence, due to mainly semantic reason.
There is another possibility that could make this sentence correct that starting early is the post-modifier in the Noun Phrase structure post modifying the Pronoun we. But start is not the kind of verb that occur in this construction.
So your sentence is grammatically wrong.
1. A Comprehensive Grammar of the English Language by Quirk et al. Page No. 1263
2. Longman Grammar of Spoken and Written English by Biber et al. Page No. 631
- ANSWER TO YOUR QUESTION 3:
The wolf wished to pick a quarrel with the lamb. He said, "How dare you make the water muddy?"
Yes you can re-write the sentence the way you said it -
The wolf wishing to pick up a quarrel with the lamb said, "How dare you make the water muddy?"
Wishing to pick up a quarrel with the lamb, the wolf said, "How dare you make the water muddy?"
The subordinate Gerund-Participle clause (FUNCTION adjunct here) - wishing to pick up a quarrel with the lamb - can express the reason of what is expressed through the matrix clause or it can express circumstantial information.
- ANSWER TO YOUR QUESTION 4:
The porter opening the gate, we entered.
In my opinion, this is grammatically correct sentence.
I am quoting similar examples from the grammar books -
An excerpt From The Cambridge Grammar of the English Language Page No. 1265
 i a. His hands gripping the door, he let out a volley of curses.
b. This done, she walked off without another word.
ii a. Realising he no longer had the premier's support, Ed submitted his resignation.
b. Born in Aberdeen, Sue had never been further south than Edinburgh.
iii Whether working or relaxing, he always has a scowl on his face.
The underlined non-finites are supplements with the main clause as
anchor. Those in [i] contain a subject, and belong to what is known as
the absolute construction, one which is subordinate in form but with
no syntactic link to the main clause. Those in [ii] have no subject,
and are syntactically related to the main clause in that the missing
subject is controlled by the subject of the main clause: it was Ed
who realised he no longer had the premier's support, and Sue who was
born in Aberdeen. In neither [i] nor [ii] is there any explicit
indication of the semantic relation between the supplement and the
anchor. This has to be inferred from the content of the clauses and/or
the context. The natural interpretation of the supplement in [ib], for
example, is temporal ("when this was done"), and of that in [iia]
causal ("because he realised ... "). Both constructions allow
gerund-participials or past-participials - and also verbless forms, as
exemplified in §1o. Example [iii] belongs to the exhaustive
conditional construction discussed in Ch.11, §5·3·5· ...
From Oxford Modern English Grammar by Bas Aarts Page No. 230
Egyptian archaeological sites being what they are, Hekekyan came across some stone object or monument almost everywhere where he put in a drill.
The book says this subordinate clause gives circumstantial information.