Heb 13:10 should not be separated from v9:

It is good for our hearts to be strengthened by grace, not by eating ceremonial foods, which is of no benefit to those who do so. (NIV)

 

it is good for the heart to be strengthened by grace, not by foods, through which those who were so occupied were not benefited. (NASB)

Thus, the author of Hebrews is making a distinction between grace and ritual eating of special foods; that the emphasis here is the contrast between grace and legalism and between the hose who accept free grace and those who try to earn salvation by works of ceremonial eating.

In v10 the author then turns this into a metaphor by an allusion to the temple/tabernacle altars which had food dedicated for certain people such as the priests and Levites (Lev 6:16-18, 7:15, 16, 31-34, Num 18:8-10, Deut 18:1, 2, see also 1 Cor 9:13). The author does this by saying, "We have an altar from which those who serve the tent have no right to eat."

That is, because we "eat" at the altar of grace, those legalists who cling to he old Mosaic system cannot eat at our altar of grace because they insist on eating at the altar of legalism. Or, in other words, while those who cling to the old ceremonial system and maintain their legalism, they will never understand Christ' grace. see also Eph 2:8-10.

Now specifically, to answer the questions:

1. "We" is the people who understand and accept Christ' grace
2. The "altar" is the altar of grace (here a metaphor)
3. "Those who serve" are those that cling to the old ceremonial system of eating ceremonial foods in order to demonstrate their righteousness, which is legalism
4. I do not quite understand Q4
5. "no right to eat" translates the word, "exousia", here meaning that one can only participate in Christ's free grace when any idea of earning salvation and righteousness is abandoned. And since those who serve have not abandoned their legalism, they cannot participate in Christ's free grace.

Heb 13:10 should not be separated from v9:

It is good for our hearts to be strengthened by grace, not by eating ceremonial foods, which is of no benefit to those who do so. (NIV)

it is good for the heart to be strengthened by grace, not by foods, through which those who were so occupied were not benefited. (NASB)

Thus, the author of Hebrews is making a distinction between grace and ritual eating of special foods; that the emphasis here is the contrast between grace and legalism and between the hose who accept free grace and those who try to earn salvation by works of ceremonial eating.

In v10 the author then turns this into a metaphor by an allusion to the temple/tabernacle altars which had food dedicated for certain people such as the priests and Levites (Lev 6:16-18, 7:15, 16, 31-34, Num 18:8-10, Deut 18:1, 2, see also 1 Cor 9:13). The author does this by saying, "We have an altar from which those who serve the tent have no right to eat."

That is, because we "eat" at the altar of grace, those legalists who cling to he old Mosaic system cannot eat at our altar of grace because they insist on eating at the altar of legalism. Or, in other words, while those who cling to the old ceremonial system and maintain their legalism, they will never understand Christ' grace. see also Eph 2:8-10.

Now specifically, to answer the questions:

1. "We" is the people who understand and accept Christ' grace
2. The "altar" is the altar of grace (here a metaphor)
3. "Those who serve" are those that cling to the old ceremonial system of eating ceremonial foods in order to demonstrate their righteousness, which is legalism
4. I do not quite understand Q4
5. "no right to eat" translates the word, "exousia", here meaning that one can only participate in Christ's free grace when any idea of earning salvation and righteousness is abandoned. And since those who serve have not abandoned their legalism, they cannot participate in Christ's free grace.