"Be careful about the government"

(2) Rabban Gamliel the son of Rabbi Yehudah HaNasi said: Excellent is the study of the Torah together with a worldly occupation; for the exertion [expended] in both of them causes sin to be forgotten. And all [study of the] Torah in the absence of a worldly occupation comes to nothing in the end and leads to sin. And all who work for the community, let them work for the [sake of the] name of Heaven; for the merit of their ancestors sustains them, And their righteousness (tsidkatam) will endure forever. And as for you [who work for the community, God says:] I credit you with a great reward, as if you [yourselves] had done it [on your own].

(3) Be careful about the government, as they approach a man only when they need him. They seem like good friends in good times, but they don't stay for him in time of his trouble.

(4) He was accustomed to say: Make His [God's] will like your will, so that He will make your will like His will. Nullify your will to His will, so that He will nullify the will of others to your will.

(10) Shemayah and Avtalyon received from them. Shemayah says, "Love work, hate lordship and do not become familiar with the government."

You who engage in the needs of the community that need to talk to the government to persuade them to act in the interest of the community, Be careful not to trust their promises and to waste the wealth of the community in distributing gifts [i.e bribes] more than necessary. Don't trust in their affection to give them information, as that may damage the public.
[What does it mean when it says the authorities appears as friends?] They appear as friends at the time the public or community leaders are giving gifts [bribes]

It is said 'be careful of the [governmental] authorities' by the same one [Rabban Shimon Ben Gamliel] who says that the one who works on behalf of the public is assisted by G-d because it means to say the governmental authorities are the opposite of the public. A powerful person because of their importance and greatness excludes themselves from being counted among the public and is differentiated from them, like the way of every person with authority even if they act as if they want to govern the public in order to fulfill the the needs of the public, that is not really the case. It appears that these two concepts, that it is that helping the public brings divine assistance and that one should be careful of the authorities is because the Misha is saying a person should exert themselves for the good of the public with all of their strengths, whatever is possible for them to do, and it may be found that a person wants to flatter and benefit the ones in power and give advice to the rulers that are against the interests of the public in order to derive benefit from their closeness to power. On this is it is said be careful of the authorities [because the] authorities do not become close to someone except to benefit themselves

(1) This 'government' (rishut) is the authorities. And in these three traits there is refinement of faithfulness and of the world. As with the absence of work, things will be tight for him and he will rob and be unfaithful. And with the pursuit of lordship, he will have challenges in the world and bad things; as since people will be jealous of him and disagree with him, he will lose his faithfulness. [It is] as they said (Sanhedrin 103b), "Once a man is appointed an officer over the community below, he becomes an evildoer above." And so [too, regarding] familiarity with the government in the early days and coming close to it, it was very unusual to escape from [the damage caused by] it in this world. As he will not pay attention to anything except that which brings him close to it. And you know the matter with Doeg - and even though the authority with which Doeg became close was the anointed of God and a prophet and God's chosen one, may He be blessed.

(1) Since Shamaya was not speaking about [things] like this. As this is a great commandment to become acquainted with them [i.e government officials] in order to supervise communal matters and Mordechai and our holy rabbi (Rabbi Yehuda HaNasi) proves [it] - Midrash Shmuel.

(It is ok to lobby officials for good of community)

Do G-ds will as if it is your will..

In the texts we use this is written on Rabban Gamliel that is mentioned before this.What is the connection between this and the other texts (about government, serving public, derech eretz?) Serving the public itself is the will of G-d who is engaged in the needs of the many, and this is the will of G-d [literally the place], because every matter of the public is his will, and the proof of this is in Brachot, "Ani Tefilati Et Ratzon, hashem et ratson" Public prayer is connected with G-d's will and a public/congregation has more existence [or permanance] and therefore one should not separate themselves from the community because that would mean one is separating oneself from the more permanent existence.