First, let's get clear about the rule. For a player to be eligible, he must be eligible both by number and position. You didn't ask about number so let's ignore that.
To be eligible by position he must be either in the backfield or on the end of the line. Four players may legally be in the backfield (because at least 7 must be on the line), and the line has 2 ends, and so 4 + 2 = 6: we will have a maximum of 6 eligible players for each down.
Now, to your specific question: if a TE is on the LOS, then he can be eligible only if he is on the end. If the WR's on his side are off the LOS, then the TE is on the end and he is eligible. If a WR is on the LOS, then the TE is not on the end of the line (the WR is), so the TE is ineligible. Note that the position of his hand is utterly irrelevant to whether he is on the end of the line.
To your other question: if you have BOTH A88 and A81 on the line, only A88 (on the end) is eligible. If either A81 or the TE go downfield (beyond the expanded NZ) and there is a pass beyond the NZ, then you have ineligible downfield and should flag it.