In the 8th panel: でも、これからはドラエモンがついてるから安心しな、おじいさん。
Those 2 から are both for giving the reason for something (because)? It's a bit confusing.
No, only the second 「から」 is for stating a reason. "Don't you worry because Doraemon will be with you from now on, Grandpa!"
「これから」 just means "from now on".
In the 5th panel I don't quite get なんだもの at the end of なにをやらせてもだめなんだもの. Is it maybe a way of emphasizing the なにをやらせてもだめ statement?
Not quite but kind of close. This 「もの」 is like a nuanced and somewhat emotional "'cuz". It is a colloquial sentence-ending particle for stating a reason with an amount of dissatisfaction towards the current situaton. In real life, it is mostly kids and women who use this 「もの」.
"'Cuz you're no good at anything."
In the 7th panel I don't get ろくなめにあわないんだ. A deconstruction or clarification of what is what in the sentence would be much appreciated. The rest I get, more or less.
It is 「ろくな[目]{め}に[遭]{あ}わないんだ。」 if one were to use kanji.
「目に遭う」 means "to experience".
「ろくな」 , all by itself, means "satisfactory" or "good", but it is often used together with a negative word as in this case -- 「ない」. It is one of those words that are often used by native speakers but not really by Japanese-learners.
「ろくな目に遭わないんだ。」= "You don't have such good experiences." ⇒ "Bad things keep happening to you."
Then, in the last panels, I don't quite understand the て after おじいさん and what he refers to by だれのこと. Is he asking who the robot means by おじいさん?
「て」＝「って」. It is a quotative particle.
『おじいさん』て、だれのこと？ = "Who do you mean by 'Grandpa'?"
