1. I dunno what you mean by "deliberate"? Yes, most banners don't have many options, but they are only including all the non-limited SSR servants, and so far there haven't been many of them.
2. The game is pretty much entirely f2p. There is no competitive element, and almost everything in the game (at least as far as the story is concerned) is doable with just 3* servants and below. On top of that, the game offers regular events that give you 4* servants at NP5 for free, and on top of that, you can always borrow strong servants from your friend list. Add to that, I see you have literally the three best supports in the game. There's practically no problem for you.
Assassin is a good banner to roll, and so is lancer, really up to you which to pick. Personally I'd go with assassin just because the servants on it are a bit better imo, though the servants in the lancer banner all have more synergy with Merlin to varying degrees.
Tamamo/Vlad aren't great with one-shot burst, they are pretty famously a sustain-focused team. Using Altera also isn't a great idea, her NP is AOE so it does less damage to a single enemy than other servants. To one-shot pillars, you need a combination of higher NP levels, suitable servants (pretty much going to want buster-focused, offensive servants with single-target NPs for this one), good CEs geared specifically toward increasing the damage of one servant (ox-demon king is particularly good for demon pillars on supports, since it gives a damage bonus to other servants in the party too). Usually one-shotting is done with servants like Heracles/Ozymandias/Quetzalcoatl, all with high bond for a higher damage multiplier during Solomon and support Merlins/Wavers to charge NP and buff damage.
As for the teams you are using, you are free to use any team you want, but the ones you have been using don't have that much synergy. Usually when building teams, there are two ways to do it. You can look at what enemy classes will be in that quest, then include servants that are strong against that class, ie using servants like Iskandar against casters. Berserkers are sought-after early on because they are a blanket class that deals strong damage against everything.
The other way is to form your own teams based on what goes well together for a particular strategy. A very common team is Jeanne D'Arc Alter + Merlin + Merlin, for example. Merlins buff Jalter's buster card effectiveness, and his NP gives stars, heal and NP gain over time. This is perfect for Jalter because she has a critical star absorb skill + critical damage buff, is relatively frail so benefits form the heal, and has her own NP charge over time, so there's your synergy. What results is Jalter almost never having her health drop below 50%, being able to fire off her NP every other turn, and that NP being buffed to be extremely strong and then Jalter dealing massive critical damage on top of that.
Another example more applicable to your case is Vlad + Tamamo + Jeanne/Tamamo. This is also effective as Vlad relies on his first skill and chaining his arts cards to gain NP since his base NP gain rate is bad, and Tamamo can help him with both these things. Tamamo also has an arts buff that will benefit Vlad's damage. Two Tamamos with Vlad is really impressive, but Jeanne is also used for her tankiness and general defensive support since Vlad is relatively frail.
3. Buster/arts crit teams run servants/craft essences that help them gain critical stars so that they can perform criticals a lot. Merlin is perfect for buster-crit because his third skill buffs both buster effectiveness and critical strength, and his NP gives you stars. 2030 is a good CE for these as it grants critical stars every turn, as well as joint recital/another ending/victor of the moon as they buff buster/arts damage and crit strength. An optimal buster-crit team would look something like this: + + .
Tldr: there is no wrong way to play the game, your time as f2p is not done as long as you don't want it to be, you do not necessarily "need" anything, especially when you already have Waver, Merlin and Tamamo, go for whichever banner, makes little difference, personally recommend assassin, though lancer works slightly better with Merlin.
Sorry if this was long af and a drain to read lol, dunno how to condense.
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