Welcome to our forum. I will be glad to comment on your questions.  Before I do, let me reassure you that none of the exposures that you have described put you at any meaningful risk for HIV and had you asked for whether you needed testing at all (for any STIs, including HIV), the answer would have been an emphatic no.  At the same time, I realize that, possibly fueled in part by guilt, some people choose to test for the reassurance that negative tests can provide. 

In answer to your specific questions:

1). Can I trust the online site like STDTestExpress.com or should directly go to doctor(scared of visiting). Would these sites give correct results?  STDTestExpress and similar services use typically used licensed laboratories for testing and their results can be trusted. Our problem with these services is not their tests but that they sometimes recommend unneeded tests and fail to provide counseling about what the test results mean.

2). Would the breast milk of the women cause any harm especially if i have mouth ulcers?

No, breast milk cannot transmit HIV for adults.  I would add that unless it was obvious, it was unlikely that your partner was lactating.  Mouth ulcers would not change this assessment.

3). Would Playing with her pose any threat if I any cut on fingers?

Nom, even with cuts on your fingers, there is no risk for HIV or other STIs.

4). know HIV 4th Gen test are reliable after 28 days as you doctors already told, just asking for re-assurance if there is no need for further testing.

If your test was 28 or more days after your event, you can be completely sure that the results will not change related to the exposure that you have described.  Consider yourself reassured.

I hope my comments are helpful.  The events you describe were no risk and there is no need for further testing for HIV or any other STI related to the events you have described.  EWH