Stan Laurel and Oliver Hardy were comedic geniuses, individually and together, and their partnership was deservedly iconic and one of the best there was. They left behind a large body of work, a vast majority of it being entertaining to classic comedy, at their best they were hilarious and their best efforts were great examples of how to do comedy without being juvenile or distasteful.
'The Lucky Dog' is most notable for being of historical interest, in that it was the first time Laurel and Hardy featured together on film and where this legendary partnership was born. That is the main reason to see it, but there is more to 'The Lucky Dog' for that to be the only reason for watching. There are some good elements here and it's fun to watch but Laurel and Hardy definitely went on to much better things when their style and such were properly settled.
It is amusing with some well-engineered slapstick, but later efforts were much funnier (to a more consistent level) and inventive. It would have benefitted from sharper timing and more wit and the predictability factor was high with easily foreseeable outcomes. The story is very flimsy and tends to be too busy and towards the end confused.
Do agree that 'The Lucky Dog' gets very messy in storytelling and coherence of the action towards the end, which makes the second half less enjoyable than the promising first. There is a sense of "yet to find its feet".
However, the two are on good form and their chemistry is great fun to watch. Laurel has more to do and has the funnier material which he sinks his teeth into, but Hardy is suitably formidable and his comic timing is just as good in his own way.
First half is very promising and where the best of the humour comes from. The best parts are well-timed and very amusing. 'The Lucky Dog' is never dull and there's a lot of energy, despite being too busy in latter stages. Technically and visually, while not exactly refined, 'The Lucky Dog' is quite good and the direction is competent.
In summary, worth the look and interesting but not Laurel and Hardy at their best