You said it yourself, witch bolt "ends if you use your action to do anything else". Using your bonus action isn't using your action.
In 5e, it's just "action" not "standard" action, and as you also said "movement" actions aren't a thing. PH 189 says that "On your turn, you can move a distance up to your speed and take one action."
If you spend that one action doing not-witch bolt, the spell ends. Anything else you do with your move, bonus action, or reaction are immaterial—unless those actions violate one of the restrictions of the spell you already listed (like using your bonus action to misty step out of line of sight).
The bless spell is not interrupted
The key to the answer is:
You briefly stop the flow of time for everyone but yourself.
For everyone else, delta t is zero.
No time passes for other creatures
Turns / rounds each last six seconds -- six seconds elapse from the beginning of someone's turn and the beginning of their next turn. From this game mechanic, we can say that the usual delta t from round to round is six seconds. Spells with duration of a minute can be modeled as being on a counter in this case: you count down one for each turn/round elapsed.
For everyone who is not the archmage, their time / turn / round counter remains where it was when time stop went into effect. Since no time is passing for them, the durations of their spells are, compared to the archmage's frame of reference, extended. The archmage's delta t keeps moving. For the other characters, they don't notice a difference since for them time stopped.
The bless spell's duration depends on the caster
In the case of bless, a concentration spell, the duration is keyed to the character who cast it, which is also the character who is concentrating on it. If it is a character who is not the archmage, their delta t is zero so their duration counter resumes on their next turn. (That is after time stop ends). Their spell still lasts a minute for them because time (and the counter) stopped for everyone but the archmage. That would provide the archmage with a few extra rounds / turns of the bless spell's magical effects.
If the archmage cast bless (from a ring of spell storing) it's duration would keep decreasing since time/turns are ticking away for the archmage.
@Medix2 asked for clarification in a comment:
If bless were cast by somebody else, but cast on the Archmage, would they benefit from it?
The answer is: yes. Nothing in the time stop spell says that it stops the effects of other spells. Bless, once it takes effect, keeps on taking effect until the spell ends (or concentration ends). Nothing in its text says otherwise.
Concentration
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends. (Basic Rules p. 79)
In plain English, you in those two sentences is the caster of a spell requiring concentration. Unless you lose concentration, the spell does not end.
Best Answer
RAW - Yes, a time stop user has both bonus action and reaction available
Emphasis mine. A turn is what let's a character do anything in a round of combat, and there's nothing in the wording of timestop to suggest that the turns you experience during time stop are unusual in any way, just that they happen one after the other
Let's see what the PHB has to say about turns:
The PHB then goes on to state that
Again, emphasis mine. Given that you're taking multiple turns through timestop, this reads that you should get your bonus action.
As for your reaction:
Admittedly, your usages of reaction actions are relatively limited within your own turn, but it's still there.