Can we write informally: That said, you can still go ahead and do it. (can we use 'that said' in this way?)

2) I read both 'that will explain' and 'that would explain' both used in present tense (like one person telling another). Which is right? For instance: He must have lied. That'll explain his actions.

1) Yes, you can say that (that [being] said). The being is often left out.

2) Both are correct.

The use of "would" gives the sentence a slightly conditional nuance whereas the use of "will" doesn't.
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1) Both 'that said' and 'having said that' are correct and can be used interchangeably. Both are used before introducing an opinion that will make what you have just said seem less strong.

Brazil have the best strikers. Having said that, The Netherlands are capable of giving a nasty knock.

2) 'Would' is a milder expression of 'will'. In your sentence, both are correct.

It will rain. (No matter what.)
It would rain. (It may or may not.)

Thanks, Ivan and Akhil. Reg. #2: He must have lied. That will explain his actions. Is it the same as saying 'that explains his actions'? Can we avoid 'will', then?
'That exlains it' will mean that the speaker at the time of speaking has understood why something has happened.
"OK! So you aren't participating because you have fractured your leg? Well, that explains it."

'That will explain' will mean that the speaker at the time of speaking hasn't understood the reason yet, but possibly at some future point of time he 'will'.
"We should go and look for him at the nearby cafe. May be that will explain his being in town."

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