in the process of doing ~ right now/at this moment
Structure
- Verb[ ている ] + ところ
In レジでお金かねを はらっているところ で、泥棒どろぼうが入はいって来きました why is the past tense (払っていた) also allowed? I admit, the past tense form came to me more naturally than the infinitive but the grammar notes make no reference as to wether it is interchangeable generally or just a special case
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Hey 
First of all, in clauses like AところB where the predicate of B is くる、いく、帰る, usually ところに・ところへ are used. ところで is also natural, but ところに still fits better. I will change the sentence to something like 「ところで財布を落としちゃった。」
When we have “When I was doing A, B happened” type of sentence AているところB and AていたところB are interchangeable.
Notice that A is actually in the past tense.
When it comes to:
ゲームをやっていたところだ。
ゲームをやっているところだ。
The former means that you are no longer playing, but the later expresses that playing the game is still going on.
I hope it clarifies it a bit,
Cheers!
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レジでお金（かね）を はらっているところ で、泥棒（どろぼう）が入（はい）って来（き）ました。
はらっているところ this happened in the past so why isnt はらっていtaところ a correct answer?
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払っていたところ is also acceptable but generally in Japanese the tense of a sentence only comes at the end.
The same can be said for levels of politeness - i.e. use plain form throughout a long sentence and stick ます on the end to make it polite.
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Hey and welcome on the community forums @Jl45 
Like @matt_in_mito said, usually in Japanese tense of the sentence is determined at the end. Like in this case, whether you use ている and ていた the meaning is the same, the interlocutor picks the tense from the context.
Though there are cases when you are forced to use certain forms, like past form before あと or
non-past before 前,
By the way, read this post:
I hope it helps,
Cheers
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Indeed it does!
いつも手伝ってくれてありがとうな！
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