Did you read Gesenius? The entry for פלח is as follows:
Chaldean to labour; hence to serve (so often in the Targg.); specially, to worship God (compare עבר); followed by an accusative and ל Dan. 3:12, seq.; 7:14, 27.
So the basic meaning of the word is “to labour”, which leads to the common meaning in the Targums which is to serve or be a servant or slave to a person or nation, which in Daniel produces the special (ie., not normal) sense of serving a god. In view of this, the translation “worship” is misleading, particularly since Daniel repeatedly differentiates between serving a god and falling down before to worship a god. There is no objection, therefore, to the argument that in the very different context of Daniel 7:14, 27, pelach means “to be subservient to” (a people that has dominion) rather than to “serve” (a god).
While pelach occurs only 10 times in the Hebrew/Aramaic Bible, it is found 431 times in the Targums. The semantic scope of the word has to be determined on the basis of the widest possible corpus. There are plenty of examples in the Targums (omitted from the biblical lexicons) where pelach is used with reference to a relationship of subservience between one people group and another.
As for the referent for the pronoun in verse 27….
1. And the kingdom and the dominion and the greatness of the kingdoms under the whole heaven
2. shall be given to the people (ʿam) of the saints of the Most High;
3. his kingdom (malkhuteh) shall be an everlasting kingdom,
4. and all dominions shall serve and obey. (Dan. 7:27 ESV modified)
As you note, there is no object in the Hebrew for “shall serve and obey” (line 4). malkhut at the beginning of line 3 has a third person masculine singular suffix, but the kingdom has been given by God to “the people of the saints of the Most High” (lines 1-2). It is, therefore, not God’s kingdom that is referred to in line 3 but the kingdom that has been given to the people (ʿam). The masculine suffix (“his”) refers to the masculine singular noun ʿam: the kingdom of this people—their rule over the nations—will be an everlasting kingdom.
To clarify the use of the pronouns in the Greek texts:
And he shall give the authority and the kingdom and the magnitude of all the kingdoms, which are under heaven, to the holy people (laōi) of the Most High, to reign over an everlasting kingdom, and all authorities will be subjected to him (autōi) and obey him (autōi) until the conclusion of the word. (LXX)
And the kingdom and the authority and the majesty of the kings which are under all the heaven is given to the saints of the Most High, and his kingdom is an everlasting kingdom, and all the rulers will serve and obey him. (Theod.)
In the LXX autōi readily refers back to the masculine single “people” (laos). Theodotion lacks laos, so perhaps the translator understood autou (“his”) to refer to the Most High (hypsistou). But as Phil pointed out, this is an interpretation of verses 13-14, so the singular pronoun is just as well required by the singular “one like a son of man”. In any case, whom the rulers and authorities serve is fundamentally determined by the fact that the kingdom has been given by God to the (people of) the saints of the Most High.