Of the five legendary fish, the Angler and the Mutated Carp are definitely the easiest to catch and with good practice won't require any special equipment other than an Iridium pole and basic bait.
The Glacierfish and Crimsonfish are much more difficult to catch due to how quickly they will jerk from the top to bottom of the scale. I have yet to catch the Legend; however, I was met with success in catching the other 4 by doing the following:
1) Craft premium bait (recipe obtained from Linus). The premium bait seemed to attract the legendary fish more frequently, which saves time spent attempting to catch them.
2) Purchase or craft a few Cork Bobbers and Trap Bobbers. The Cork Bobber (750g or 10 wood, 5 hardwood, 10 slime) expands the fishing bar. The Trap Bobber (500g or 1 copper bar, 10 sap) slows down how quickly the fish escape while not being reeled in. The bobbers will degrade, and having 2 of each will allow you to do a full day of fishing with no problem.
3) If your fishing level is not already 10, purchase or prepare skill boosting meals! The following meals will help you boost up your skill to 10, which expands your fishing bar as well.
Trout Soup : Fishing +1*
Chowder : Fishing +1
Escargot : Fishing +2
Fish Taco : Fishing +2
Dish O' The Sea : Fishing +3
Fish Stew : Fishing +3
Lobster Bisque : Fishing +3
*: Trout Soup is only 250G per meal from Willy's shop, so consider this an easy meal to bump from 9 to 10 if you're about to lose out on the season.
4) Keep trying and don't give up! If you're really tight on time, find a day with the perfect conditions (no crops to tend to or other responsibilities), and at the worst you can always 'Quit to Desktop' before the day is over to give it another day's attempts.
Other Advice:
For fish that like to jerk around in the water, always try to just barely have them caught. That way, when they dive up or down, you can react much more quickly and don't have to move as far.
DO NOT attempt to use the Barbed Hook on the Crimsonfish, Glacierfish, or Legend. You'll be spinning your wheels with how sporadically the fishing bar will 'adjust' -- (sway wildly).
There's no danger, no. Trees outside your farm will regrow over time. (I haven't timed it, but my offhand impression is it takes about half a season or a bit more.)
This makes town and forest trees a renewable source of wood, sap, and tree seeds. Seeds being a renewable resource means you can always reforest your farm if desired.
Best Answer
No it isn't dangerous. The only effect of storms AFAIK is to charge up Battery packs if you have lightning rods placed. Otherwise it's considered as normal rain for fishing.