1. Ordinary conversations in all languages are peppered with tags, such as 'you know', 'isn't it?' etc. Welsh is no different. When Brian tries to reassure Agnes that although Rhodri has gone missing from school all will be well, he says

Fydd pob dim yn iawn, 'sti.
Everything will be all right, you know.

'sti is an abbreviation of wyddost ti. It's the present tense of gwybod 'to know'.
So - wyddost ti, or the more usual form, 'sti' means 'you know' and is more usual in North Wales, where you'll hear it a lot!

2. When Jac says there's no point in Brian rushing off to see Rhodri, Brian retorts angrily by saying that of course he must go and see him as Rhodri is his son:

Wrth gwrs fod pwynt! Mae o'n fab i mi!
Of course there's a point. He's my son!

You'll have learnt the phrase fy mab i to translate 'my son'. But here, Brian is emphasising the fact that Rhodri's his son. Therefore he says 'Mae o'n fab i mi. Literally 'He's a son to me'.

3. Agnes informs Brian that Rhodri wants to go and live with him.
Mae'n mynnu ei fod e'n cael dod i fyw atat ti.
He insists that he's allowed to come and live with you.

The verb cael has several meanings and usages. One of them is 'to be allowed'.

Dw i'n cael aros efo Mari nos fory
I'm allowed to stay with Mari tomorrow night

It's commonly used to ask permission too. If you want to ask permission to go out, you may say:

Ydw i'n cael mynd allan heno?
Am I allowed to go out tonight?

However in this context, you're more likely to hear the short form Ga' i? (Gaf i?) which means 'may I?' rather than the long-winded Ydw i'n cael?

So Ga'i fynd allan means May I go out?

Ga'i gadw fo tan fory? means May I keep it until tomorrow?