And and With, Formal and Informal

1. And, With

When do you use 같이 (together), and how do you use it in a sentence? I was looking for example sentences on Tatoeba.

I know that to indicate doing something "with a friend" is 친구하고 .

Example: 친구하고 먹었어요. (I ate with a friend.)

But we first learnt that 하고 means "and" for connecting nouns. And the formal version of 하고 is 와/과 (depending on whether there is a Batchim; this is the one with the "inverse" rule).

I looked at Tatoeba and on the first page of results saw this sentence: 친구와 같이 텔레비전에서 축구시합을 봅니다.

Translation: I watch a football match on television with my friend. (together with my friend?)

So the question in my mind was: Does it also mean that the formal "와/과" for "and" also means "with", but the formal form?

2. Formal, Informal

Being more studious now (after that wake-up call last lesson), I have been reading up more on my own.

According to How to Study Korean, this is what it says about the first person singular pronoun:

저 = I, me (formal)
나 = I, me (informal)

I've not learnt 나 in class yet. But reading the textbook while trying to do my homework, I realise it was used in one sample journal entry.

Is it weird that for our sentences they start with formal 저(는), but then have a verb that ends with the polite casual/informal -요? (The tense is called 해요체 according to Wiktionary.)

I had this question before when we learnt about 하고 and 와/과 with the meaning "and".

I found it weird that sentences using the formal 와/과 would end with -요 instead of the formal polite tense (하십시오체), that ends with -ᆸ니다 (at least in the imperative, according to Wiktionary again).

Does this notion of formal/informal not have to be consistent between the nouns/conjunctions and the verbs? Are they a different dimension of formality? Or is it because this system is complex that they don't burden beginners with it?