How to Choose Support Servants?

Pretty sure it would be waver followed by archuria with Drake being a very distant third for most people. Personally though I generally stick support servants in the 6th slot and just use them for CEs.

Outside of events just put the highest leveled and skilled servants you can in each slot. Except for the berserker slot. Fran or Lancelot is best there with kaleidoscope as AOE farmers

Agree, just stick your highest leveled, highest skill servants in there with the exception for the berserker slot.

For me though I'd like to see Jekyll just because that's a servant I never see on people's supports and I think he's cool xD

Use whoever you like, I know some people that have wavers and instead put nursery rhyme or medea as their support. Ideally tho, you should choose the ones higher in the tier list and level them up as fast as you can

If you're referring to your support lineup for your friends to use, pick the best out of each of the classes you have. YMMV, but here's who I'd pick out of your current lineup:

Saber: Artoria or Lancelot, since the latter is a really good ST while Artoria tends to be pretty generic, unfortunately, despite being the flagship of the whole bloody series.
Archer: Squirtoria
Lancer: Brynhild
Rider: Drake
Caster: Waver (duh lol)
Assassin: Shiki (I see yours is level 90. A man of exquisite taste, you are.)
Berserker: Vlad

As for events, place whichever provides event-related bonuses. For events like summer, make sure to vary your lineup so people have options, preferably matching the proper servant with the associated bonus, like if a certain fight would be best with Lancers and you have a Lancer that provides a bonus to that quest's currency drop, put him/her in there. If you don't have any event bonus servants for the class, just put your strongest in, same as non-event lineup.

You've got a solid lineup and don't really need support so some strategies can be
1) grab whomever and toss it in the back.
2) grab someone you don't have and want to test out.
3) if it is too expensive to field all your desired ssr and 5* event ce's, grab a support ssr (who is free) of who you want, then pick a lower tier of your own.
4) play with someone you wish you had but the gatcha gods still deny you.

I'm no expert at this game, but if I were to say which of your servants I'd prefer to use as a support from each class:

Altria (if you have Kaleidoscope on her, otherwise probably Lancelot), Archuria, Bryn, Drake, Waver, Shiki, Lancelot/Fran (if you have Kaleidoscope)

Out of the servants you have this is what I recommend:
Saber slot: Lancelot, great arts servant, I’d take him over Artoria any day
Archer slot: Archuria may seem to be the obvious choice, but personally, I don’t see that many people use Tesla, it’d really make your support roster stand out.
Lancer: Bryn, like Tesla, not many people have her on their support and in my experience she has garnered me a good amount of friend points the past few events. People really want to use her so definitely give them that chance.
Rider: Drake, she’s such a top tier farmer.
Caster:Waver or Tamamo will give you all the friend requests in the world. Maybe think about putting one of them in your all slot
Assassin: Shiki, the fact that you grailed her will make her stand out compared to most people who have her on their support.
Berserker: Lancelot again, like Drake he is a top tier farmer.