ZDC Center Controller reported there were coordination issues with other facilities around the area that caused delays in handoffs; leading to airborne conflicts and overall unsafe operations.

Narrative

Training in progress at Sector 58/59 combined. Sector 12 had a handoff pending (Aircraft X) to our sector; and the trainee did not take it in a timely manner. Obviously; Sector 12 is required to either keep the aircraft out of our airspace; or complete the radar handoff. The aircraft entered our airspace without coordination; continuing for over 15 miles and more than 2 full minutes before they finally called.Aircraft X was in clear and obvious conflict with Aircraft Y in our airspace (FL280 climbing to FL340); with about 2 minutes until separation would definitely have been lost. As the On-The-Job Training Instructor; I had recently taken over the frequency; with the R58/59 trainee continuing to handle the handoffs and typing. I stopped our traffic (Aircraft Y) at FL320 to resolve the conflict; but took no other action. The trainee still did not notice the handoff and it was still not being accepted; but Sector 12 continued to not do anything about it. They eventually called us; to which I responded with frustration about the second such error being made by that area within an hour; and during this exchange the R58/59 trainee then recognized the situation and accepted the handoff while Sector 12 and I discussed the issue. Both sides were relatively unprofessional on the line. The On-The-Job Training Instructor at Sector 12 said that if there was an issue; to 'give us a call then' - but that's nonsense. They are responsible to accomplish the handoff prior to the airspace boundary; and the implication that we; as the receiving sector; should just get violated and accept the blame for being violated; makes no sense. It does however very much seem to be the widespread opinion in that area; which is my primary concern; rather than the specifics of any one single occurrence.To reinforce the clear presence of this poor attitude; Sector 12 retaliated against my attitude in our previous call by calling us on the next handoff at 20 NM (~2.5 minutes) from the boundary; around triple the norm; presumably just to make a point. The aircraft was not visible (no display) due to the range; but upon ranging out; I was able to find them; move the data block; and accept the handoff. Knowing the two people involved; the R side developmental at 12 was almost certainly doing this only because the On-The-Job Training Instructor involved explicitly directed him to; with purely retaliatory intent. It's pretty bold to continue to go on the attack after committing an airspace violation; and that attitude speaks to the way I'm claiming that they are not putting safety and separation first with handoffs to Sector 58.We should not be in situations where the receiving sector must take evasive action to protect traffic in our own airspace from traffic inbound from an adjacent sector who does not have a completed radar handoff. This was one of two nearly identical situations with the same area within the same hour today (different sector).Unclear what to do about it; but it is a clear pattern that Area 1 is consistently late to call on handoffs in situations like these. They seem to treat it as a formality; and only call once there is nothing that could possibly be done to avert any conflicts that may have been developing; unless we take action as the receiving sector to move our own traffic preemptively; since we know they will not evaluate conflicts on our shared boundary; and will also not call in a timely manner to bring our attention to the handoff. It is my opinion that traffic from Sectors 12/19 entering Sector 58 are not treated with the appropriate level of precaution by Area 1's controllers; despite the nature of the perpendicular crossing paths with climbing traffic; and they need to call with enough time to actually do something; rather than only bringing it to our attention 15 - 30 seconds from the boundary (at which point we basically have no choice but to accept and move our own traffic; in our own airspace; to accommodate their lackof precaution/foresight). The attitude of 'it works in my airspace' and 'I called you on the handoff before the boundary' is not good enough to ensure a safe operation.