Are there any types of weekly activities that are bugged or not worth doing, that should be avoided at the moment?
Most of the weekly activity types are absolutely fine [1] if you get the generic "do this activity type in this system type" and have a fair chance of being impossible if they're "do this activity type in this specific system".
As a workaround, if you make sure to finish each week / start each new week well clear of your own power's space, you should only get the generic ones; at the very least try not to finish the week so deep in your own Power's space that there's no enemy systems nearby, as that makes a lot of the "specific system" bugs more likely.
[1] Once you've got the hang of the various bits of terminology you need to understand a request like "deliver 20t of Power Commodities to an Acquisition System", that is.
On top of the weekly activities, anything else worth doing?
Pretty much
any normal activity done in your own Power's territory will get you some merits, either directly or as a side-effect. (Depending on your luck with getting easy weekly tasks, this can be faster than the weekly tasks are).
My personal preference is to fly a multi-functional ship with cargo space, Operations Limpet multi-controller, and a good weapons/shields set up - that will be capable of doing several different powerplay tasks without a refit, which means you can just take opportunities for a few merits as they show up in normal play.
I am not really interested in the perks since I have billions in the bank...but are there some Powers more "fun" than others?
There are a couple of non-financial perks which might be of interest anyway:
- Kaine (the strongest for this), Winters and Aisling Duval give accelerated reputation gain with minor factions in their respective territories, which can be useful if you like doing missions but also move around a bit
- Delaine (uniquely) gives bounty nullification in his territory.
In terms of merit earning, all Powers have their set of activities which their prefer - those get a 50% earning bonus. Broadly speaking, these break down into five "combat" powers (Delaine, Archer, Lavigny-Duval, Grom, Patreus) and seven "non-combat" powers. So you'll rank up faster - and also score control points for your Power's systems faster - by picking a power with activities that match your own preferred activities.
If you have a particular part of the bubble you prefer to play in, for whatever reason, then picking a power with some presence in that region is a good idea - the weekly assignments have a tendency towards being "Reinforcement" tasks where you strengthen your own side's systems
Or, any of them somehow increasing the chance to meet other players (friendly or unfriendly, both are good to me!)
Any power should work for that (on the friendly side there'll be various power-aligned groups, some with more-or-less open priority lists).
For unfriendly players you want a power which is getting involved in various flashpoints rather than just sitting back and reinforcing; the Federal powers' central position means that someone is always attacking them, so they're probably the ones with the best chance of seeing action. Whether you pick a Federal power or one of their enemies is more a matter of taste.
I'm pledged to Delaine and I'm having a blast.