and pronounced as /ənd/ here? Or /ænd/?

ænd

http://oald8.oxfordlearnersdictionaries.com/dictionary/and

I can't do all your errands.

while my cousins in the UK would say, al-yoo-min-ee-um.

Ori, you have to understand that the various accents in the English speaking word are often based on vowels. Where I might hold a vowel, or soften it, another might make it a hard. While I may pronounce a word as the dictionary describes, someone from another area may not. Take the word alumin(i)um. I would pronounce it as al-oo-min-um, while my cousins in the UK would say, al-yoo-min-ee-um.
So, while I might say and like band, some people out there might pronounce it otherwise. Again, they wouldn't vary in the way they say it either.

There ISN'T a Standard English pronunciation. There are a bunch. In America, normal midwestern English would be about the closest--that's like what most broadcasters, actors, and such are expected to speak, for the most part. But we have dozens of regional accents. Britain does too, tho maybe BBC-speak would be considered a standard. Maybe Monty Python-speak would be sort of a standard, when they weren't doing accents.

I'd say most people in general conversation, when they aren't paying particular attention to their speech, would usually elide "and" to just 'n', a little more attention might get
"(schwa)nd" and if you were going to emphasize it you'd get "aend" (I don't remember how to get to the phonetic symbols keyboard). You probably (wouldn't stand out particularly if you used any of the three. (I'm using 'n' as sort of very-weak-schwa-n, NOT pronounced like the letter n).

Ori, you will always have an accent. Always. I have one. Everyone does. Some are just harder to pinpoint.
There are certain sounds that are difficult for people mimic. Chinese seem to struggle with L, Italians with TH like with or bath, Hungarian and Danes pronounce W as V.
The kids I went to school with all had canadian tinged accents when they spoke their parents language. When my fully fluent daughter went to France and Quebec, they knew she was from away.
Phonetics are a guideline, that's it.