[RPG] My DM has removed heavy armour from the game. How to make a Paladin work or convince him not to do this

You're right. There is no clear cut way. That's by design. How this works varies a lot by playstyle:

Some groups have a solid understanding of ethics and jurisprudence and use some limited form of said understanding adjudicate alignment in D&D. Others use their full understanding of ethics, which cumulates for them into a D&D compatible system, to perform that adjudication. These groups are very similar except that the first group tends to have less issues adjudicating alignment in difficult situations where the latter group discovers holes in their beliefs about morality. In both of these cases alignment is adjudicated according to the chosen moral/ethical system, and what constitutes oath-breaking will be adjudicated similarly according to the related principles of jurisprudence. As moral relativism becomes, unfortunately, increasingly popular, objective alignment becomes less and less so, especially amongst simulationist groups, and so these two methods of alignment adjudication are unlikely.

Some groups think alignment is stupid, meaningless, and entirely subjective. These people are very vocal. They likely also think that oathbreaking is similarly problematic. Like alignment-based powers in previous editions, in these groups if you're going to use anything that's based off alignment or alignment-like systems (like the paladin class) you should first check with your DM to find out 1) if it's allowed and 2) what it means in the context of the campaign. One common solution in the past in these kinds of groups has been just to waive the alignment restriction entirely, for the Paladin class.

Some groups think that obviously alignment is objective, because it's in the rules, but they just don't understand it yet. These groups will post difficult alignment questions as they come up on online fora. The outcome of alignment based actions in such groups will be fickle and likely a point of tension.

Some groups try and adjudicate this according to the original inspiration for the alignment system, which is a set of novels written by Michael Moorcock. This is hard because the novels are novel-y and don't explain exactly what Law and Chaos are in cut and dry terms. Such groups would probably not adjudicate the oath the same way they do alignment, because Jurisprudence is fundamentally a principle of Balance, not Law, in that system, and there's almost nothing to go on for how something like the 5e paladin oath should be dealt with.

In any case, if you are not the DM, ask your DM how the metaphysics of ethics and jurisprudence are related in their game. If they are related, seek to focus more on the adjudication of ethics than jurisprudence as the ethical component is probably dominant in most GM adjudications. If they are not related, ask about Jurisprudence in general, and the adjudication of your class abilities in particular, if necessary, focusing on what an Oath means, what it means to break it, and how reparations might be made if the oath is indeed broken.

If you are the DM, I strongly recommend you consciously decide what philosophical system of justice you will use in your campaign, as all of the other options amount to unconsciously using some system and putting some introspective thought into this choice before you start running the game is likely to make these kinds of decisions much easier and much faster during the game sessions, since you will spend much less time figuring out how to approach each specific case.

If you want a place to start looking at different approaches to the philosophy of oaths, I recommend this paper.

Most background NPCs are defined by the setting

The right answer is going to vary significantly based on the personality of your party. The party I DM is rather incurious, but I play in a super curious party, so I'll try to address both areas.

The vast majority of people in a city are going to be boring (at least from a PC's point of view). Merchants, day laborers, layabouts--all of these people aren't going to be doing anything particularly complex or interesting. They are likely to be indifferent to the PCs, and so won't sustain long conversations.

You can flesh out these background characters by fleshing out your city. If your city is a mercantile port city, for example, you can make sailors and merchants. If that city deals mostly in, say, rare ocean gems, you can include pirates and mercenaries, as well as jewelers. The roleplay of these characters thus comes from the setting, rather than individually defined NPCS, allowing you to mix and match aspects for individuals.

Have a handful of interesting NPCs

I like to come up with maybe 2-4 NPCs that are actually interesting. Perhaps the PCs might meet a pre-necromancer Vecna, or a contact for a magic items dealer. These characters have some particularly significant backstory and are potentially story hooks. The party is unlikely to actually interact with that many NPCs (how many people actually talk to each other on the streets of NYC?), and these interesting NPCs are a kind of reward for exploring.

Define NPCs by motivation and disposition

Like you, I'm terrible at accents and only passable at actually portraying different characters. I try to vary the way my NPCs talk (smart, dumb, etc.), but it's difficult to do on the fly. The way I keep my NPCs distinct is by significantly varying their dispositions and motivations. Maybe one contact they have to meet is super suspicious of the party, and constantly acts paranoid. Maybe another is obsequious toward the party, because he wants to scam them.

Distinguishing NPCs this way is quick and easy. For example, Archie is a poor merchant looking for his next big break, and sucks up to the PCs, whereas Bart is a surly guard who thinks the PCs are up to no good. Such short bios are easy to make up on the fly, and switching between antagonistic NPCs and nice NPCs is an easy way to help the PCs distinguish between them.